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{{wiktionary|Appendix:Glossary of topology}}

This is a glossary of some terms used in the branch of [[mathematics]] known as [[topology]]. Although there is no absolute distinction between different areas of topology, the focus here is on [[general topology]]. The following definitions are also fundamental to [[algebraic topology]], [[differential topology]] and [[geometric topology]]. For a list of terms specific to algebraic topology, see [[Glossary of algebraic topology]].

All spaces in this glossary are assumed to be [[topological space]]s unless stated otherwise.

{{Compact ToC|short1|sym=no|seealso=yes|refs=yes}}

== A ==

;Absolutely closed: See ''H-closed''

;Accessible: See '''[[T1 space|<math>T_1</math>]]'''.

;Accumulation point: See [[limit point]].

;[[Alexandrov topology]]:  The topology of a space ''X'' is an [[Alexandrov topology]] (or is '''finitely generated''') if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the [[upper set]]s of a [[poset]].<ref>Vickers (1989) p.22</ref>

;Almost discrete: A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.

;α-closed, α-open: A subset ''A'' of a topological space ''X'' is α-open if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X \left( \operatorname{Int}_X A \right) \right)</math>, and the complement of such a set is α-closed.{{sfn|Hart|Nagata|Vaughan|2004|p=9}}

;[[Approach space]]:  An [[approach space]] is a generalization of metric space based on point-to-set distances, instead of point-to-point.

== B ==

;Baire space: This has two distinct common meanings:
:#A space is a '''Baire space''' if the intersection of any [[countable]] collection of dense open sets is dense; see [[Baire space]].
:#'''Baire space''' is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see [[Baire space (set theory)]].

;[[Base (topology)|Base]]: A collection ''B'' of open sets is a [[base (topology)|base]] (or '''basis''') for a topology <math>\tau</math> if every open set in <math>\tau</math> is a union of sets in <math> B </math>. The topology <math>\tau</math> is the smallest topology on <math>X</math> containing <math>B</math> and is said to be generated by <math>B</math>.

;[[Basis (topology)|Basis]]: See '''[[Base (topology)|Base]]'''.

;β-open: See ''Semi-preopen''.

;b-open, b-closed: A subset <math>A</math> of a topological space <math>X</math> is b-open if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X A \right) \cup \operatorname{Cl}_X \left( \operatorname{Int}_X A \right)</math>. The complement of a b-open set is b-closed.{{sfn|Hart|Nagata|Vaughan|2004|p=9}}

;[[Borel algebra]]: The [[Borel algebra]] on a topological space <math> (X,\tau)</math> is the smallest [[Sigma-algebra|<math>\sigma</math>-algebra]] containing all the open sets. It is obtained by taking intersection of all <math>\sigma</math>-algebras on <math> X </math> containing <math> \tau </math>.

;Borel set: A Borel set is an element of a Borel algebra.

;[[Boundary (topology)|Boundary]]: The [[boundary (topology)|boundary]] (or '''frontier''') of a set is the set's closure minus its interior.  Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set <math> A </math> is denoted by <math> \partial A</math> or <math>bd</math> <math>A</math>.

;[[Bounded set|Bounded]]: A set in a metric space is [[bounded set|bounded]] if it has [[finite set|finite]] diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A [[function (mathematics)|function]] taking values in a metric space is [[bounded function|bounded]] if its [[image (functions)|image]] is a bounded set.

== C ==

;[[Category of topological spaces]]: The [[category theory|category]] '''[[Category of topological spaces|Top]]''' has [[topological space]]s as [[object (category theory)|objects]] and [[continuous map]]s as [[morphism]]s.

;[[Cauchy sequence]]: A [[sequence]] {''x''<sub>''n''</sub>} in a metric space (''M'', ''d'') is a [[Cauchy sequence]] if, for every [[positive number|positive]] [[real number]] ''r'', there is an [[integer]] ''N'' such that for all integers ''m'', ''n'' > ''N'', we have ''d''(''x''<sub>''m''</sub>, ''x''<sub>''n''</sub>) < ''r''.

;[[Clopen set]]: A set is [[clopen set|clopen]] if it is both open and closed.

;Closed ball: If (''M'', ''d'') is a [[metric space]], a closed ball is a set of the form ''D''(''x''; ''r'') := {''y'' in ''M'' : ''d''(''x'', ''y'') ≤ ''r''}, where ''x'' is in ''M'' and ''r'' is a [[positive number|positive]] [[real number]], the '''radius''' of the ball. A closed ball of radius ''r'' is a '''closed ''r''-ball'''. Every closed ball is a closed set in the topology induced on ''M'' by ''d''. Note that the closed ball ''D''(''x''; ''r'') might not be equal to the [[closure (topology)|closure]] of the open ball ''B''(''x''; ''r'').

;[[Closed set]]: A set is [[Closed set|closed]] if its complement is a member of the topology.

;[[Closed function]]: A function from one space to another is closed if the [[image (mathematics)|image]] of every closed set is closed.

;[[Closure (topology)|Closure]]: The [[closure (topology)|closure]] of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set ''S'' is a '''point of closure''' of ''S''.

;Closure operator: See '''[[Kuratowski closure axioms]]'''.

;[[Coarser topology]]: If ''X'' is a set, and if ''T''<sub>1</sub> and ''T''<sub>2</sub> are topologies on ''X'', then ''T''<sub>1</sub> is [[coarser topology|coarser]] (or '''smaller''', '''weaker''') than ''T''<sub>2</sub> if ''T''<sub>1</sub> is contained in ''T''<sub>2</sub>. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''stronger'''.

;Comeagre: A subset ''A'' of a space ''X'' is '''comeagre''' ('''comeager''') if its [[complement (set theory)|complement]] ''X''\''A'' is [[meagre set|meagre]]. Also called '''residual'''.

;[[Compact space|Compact]]: A space is [[compact space|compact]] if every open cover has a [[finite set|finite]] subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact [[Hausdorff space]] is normal. See also '''quasicompact'''.

;[[Compact-open topology]]: The [[compact-open topology]] on the set ''C''(''X'', ''Y'') of all continuous maps between two spaces ''X'' and ''Y'' is defined as follows: given a compact subset ''K'' of ''X'' and an open subset ''U'' of ''Y'', let ''V''(''K'', ''U'') denote the set of all maps ''f'' in ''C''(''X'', ''Y'') such that ''f''(''K'') is contained in ''U''. Then the collection of all such ''V''(''K'', ''U'') is a subbase for the compact-open topology.

;[[Complete space|Complete]]: A metric space is [[complete space|complete]] if every Cauchy sequence converges.

;Completely metrizable/completely metrisable: See '''[[complete space]]'''.

;Completely normal: A space is completely normal if any two separated sets have [[Disjoint sets|disjoint]] neighbourhoods.

;Completely normal Hausdorff: A completely normal Hausdorff space (or [[T5 space|'''T<sub>5</sub>''' space]]) is a completely normal T<sub>1</sub> space. (A completely normal space is Hausdorff [[if and only if]] it is T<sub>1</sub>, so the terminology is [[consistent]].) Every completely normal Hausdorff space is normal Hausdorff.

;[[Completely regular space|Completely regular]]: A space is [[Completely regular space|completely regular]] if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and {''x''} are functionally separated.

;[[Completely T3 space|Completely T<sub>3</sub>]]: See '''[[Tychonoff space|Tychonoff]]'''.

;Component: See '''[[Connected space|Connected component]]'''/'''Path-connected component'''.

;[[Connected (topology)|Connected]]: A space is [[connected (topology)|connected]] if it is not the union of a pair of [[Disjoint sets|disjoint]] nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.

;[[connected space|Connected component]]: A [[connected space|connected component]] of a space is a [[maximal set|maximal]] nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a [[partition of a set|partition]] of that space.

;[[Continuity (topology)|Continuous]]: A function from one space to another is [[continuity (topology)|continuous]] if the [[preimage]] of every open set is open.

;[[Continuum (topology)|Continuum]]: A space is called a continuum if it a compact, connected Hausdorff space.

;[[Contractible space|Contractible]]: A space ''X'' is contractible if the [[identity function|identity map]] on ''X'' is homotopic to a constant map. Every contractible space is simply connected.

;[[Coproduct topology]]: If {''X''<sub>''i''</sub>} is a collection of spaces and ''X'' is the (set-theoretic) [[disjoint union]] of {''X''<sub>''i''</sub>}, then the coproduct topology (or '''disjoint union topology''', '''topological sum''' of the ''X''<sub>''i''</sub>) on ''X'' is the finest topology for which all the injection maps are continuous.

;[[Core-compact space]]

;[[Cosmic space]]: A [[continuous function|continuous]] [[Image (mathematics)|image]] of some [[Separable space|separable]] [[metric space]].<ref>{{cite book | title=Encyclopedia of Distances | first1=Michel Marie | last1=Deza | first2=Elena | last2=Deza |author2-link=Elena Deza| publisher=[[Springer-Verlag]] | year=2012 | isbn=978-3642309588 | page=64 }}</ref>

;[[Countable chain condition]]: A space ''X'' satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.

;[[Countably compact]]: A space is countably compact if every [[countable]] open cover has a [[finite set|finite]] subcover. Every countably compact space is pseudocompact and weakly countably compact.

;Countably locally finite:  A collection of subsets of a space ''X'' is '''countably locally finite''' (or '''σ-locally finite''') if it is the union of a [[countable]] collection of locally finite collections of subsets of ''X''.

;[[Cover (topology)|Cover]]: A collection of subsets of a space is a cover (or '''covering''') of that space if the union of the collection is the whole space.

;Covering: See '''Cover'''.

;Cut point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a cut point if the subspace ''X'' − {''x''} is disconnected.

== D ==
;δ-cluster point, δ-closed, δ-open: A point ''x'' of a topological space ''X'' is a δ-cluster point of a subset ''A'' if <math>A \cap \operatorname{Int}_X\left( \operatorname{Cl}_X(U) \right) \neq \emptyset</math> for every open neighborhood ''U'' of ''x'' in ''X''.  The subset ''A'' is δ-closed if it is equal to the set of its δ-cluster points, and δ-open if its complement is δ-closed.{{sfn|Hart|Nagata|Vaughan|2004|pp=8–9}}

;[[Dense set]]: A set is dense if it has nonempty intersection with every nonempty open set. Equivalently, a set is dense if its closure is the whole space.

;[[Dense-in-itself]] set: A set is dense-in-itself if it has no [[isolated point]].

;Density: the minimal cardinality of a dense subset of a topological space.  A set of density ℵ<sub>0</sub> is a [[separable space]].<ref>Nagata (1985) p.104</ref>

;Derived set: If ''X'' is a space and ''S'' is a subset of ''X'', the derived set of ''S'' in ''X'' is the set of limit points of ''S'' in ''X''.

;Developable space: A topological space with a [[Development (topology)|development]].<ref name=ss163/>

;[[Development (topology)|Development]]: A [[countable set|countable]] collection of [[open cover]]s of a topological space, such that for any closed set ''C'' and any point ''p'' in its complement there exists a cover in the collection such that every neighbourhood of ''p'' in the cover is [[disjoint sets|disjoint]] from ''C''.<ref name=ss163/>

;Diameter: If (''M'', ''d'') is a metric space and ''S'' is a subset of ''M'', the diameter of ''S'' is the [[supremum]] of the distances ''d''(''x'', ''y''), where ''x'' and ''y'' range over ''S''.

;Discrete metric: The discrete metric on a set ''X'' is the function ''d'' : ''X'' × ''X'' &nbsp;→&nbsp; '''[[real number|R]]''' such that for all ''x'', ''y'' in ''X'', ''d''(''x'', ''x'') = 0 and ''d''(''x'', ''y'') = 1 if ''x'' ≠ ''y''. The discrete metric induces the discrete topology on ''X''.

;[[Discrete space]]: A space ''X'' is [[discrete space|discrete]] if every subset of ''X'' is open. We say that ''X'' carries the '''discrete topology'''.<ref name=ss41>Steen &amp; Seebach (1978) p.41</ref>

;[[Discrete topology]]: See '''[[discrete space]]'''.

;Disjoint union topology: See '''Coproduct topology'''.

;[[Dispersion point]]: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a dispersion point if the subspace ''X'' − {''x''} is hereditarily disconnected (its only connected components are the one-point sets).

;Distance: See '''[[metric space]]'''.

;[[Dowker space]]

;[[Dunce hat (topology)]]

== E ==

;[[Entourage (topology)|Entourage]]: See '''[[Uniform space]]'''.

;Exterior: The exterior of a set is the interior of its complement.

== F ==

;[[F-sigma set|''F''<sub>σ</sub> set]]: An [[F-sigma set|''F''<sub>σ</sub> set]] is a [[countable]] union of closed sets.<ref name=ss162/>

;[[Filter (set theory)|Filter]]: See also: [[Filters in topology]]. A filter on a space ''X'' is a nonempty family ''F'' of subsets of ''X'' such that the following conditions hold:
:# The [[empty set]] is not in ''F''.
:# The intersection of any [[finite set|finite]] number of elements of ''F'' is again in ''F''.
:# If ''A'' is in ''F'' and if ''B'' contains ''A'', then ''B'' is in ''F''.

;[[Final topology]]: On a set ''X'' with respect to a family of functions into <math>X</math>, is the [[finest topology]] on ''X'' which makes those functions [[continuous function (topology)|continuous]].<ref>{{cite book | first=Stephen | last=Willard | title=General Topology | url=https://archive.org/details/generaltopology00will_0 | url-access=registration | year=1970 | publisher=Addison-Wesley | location=Reading, MA | zbl=0205.26601 | series=Addison-Wesley Series in Mathematics| isbn=9780201087079 }}</ref>

;[[Fine topology (potential theory)]]: On [[Euclidean space]] <math>\R^n</math>, the coarsest topology making all [[subharmonic function]]s (equivalently all superharmonic functions) continuous.<ref>{{cite book | first=John B. | last=Conway | author-link=John B. Conway | isbn=0-387-94460-5 | series=[[Graduate Texts in Mathematics]] | volume=159 | title=Functions of One Complex Variable II | publisher=[[Springer-Verlag]] | year=1995 | zbl=0887.30003 | pages=367–376 }}</ref>

;[[Finer topology]]: If ''X'' is a set, and if ''T''<sub>1</sub> and ''T''<sub>2</sub> are topologies on ''X'', then ''T''<sub>2</sub> is [[finer topology|finer]] (or '''larger''', '''stronger''') than ''T''<sub>1</sub> if ''T''<sub>2</sub> contains ''T''<sub>1</sub>. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''weaker'''.

;Finitely generated: See '''[[Alexandrov topology]]'''.

;[[First category]]: See '''[[Meagre set|Meagre]]'''.

;[[First-countable space|First-countable]]: A space is [[First-countable space|first-countable]] if every point has a [[countable]] local base.

;Fréchet: See '''[[T1 space|T<sub>1</sub>]]'''.

;Frontier: See '''[[Boundary (topology)|Boundary]]'''.

;Full set: A [[compact space|compact]] subset ''K'' of the [[complex plane]] is called '''full''' if its [[complement (set theory)|complement]] is connected. For example, the [[closed unit disk]] is full, while the [[unit circle]] is not.

;Functionally separated: Two sets ''A'' and ''B'' in a space ''X'' are functionally separated if there is a continuous map ''f'': ''X'' &nbsp;→&nbsp; [0, 1] such that ''f''(''A'') = 0 and ''f''(''B'') = 1.

== G ==

;[[G-delta set|''G''<sub>δ</sub> set]]: A [[G-delta set|''G''<sub>δ</sub> set]] or '''inner limiting set''' is a [[countable]] intersection of open sets.<ref name=ss162>Steen &amp; Seebach (1978) p.162</ref>

;''G''<sub>δ</sub> space: A space in which every closed set is a ''G''<sub>δ</sub> set.<ref name=ss162/>

;[[Generic point]]: A [[generic point]] for a closed set is a point for which the closed set is the closure of the singleton set containing that point.<ref>Vickers (1989) p.65</ref>

== H ==

; [[Hausdorff space|Hausdorff]]: A [[Hausdorff space]] (or '''[[T2 space|T<sub>2</sub>]] space''') is one in which every two distinct points have [[Disjoint sets|disjoint]] neighbourhoods. Every Hausdorff space is T<sub>1</sub>.
; [[H-closed space|H-closed]]: A space is H-closed, or '''Hausdorff closed''' or '''absolutely closed''', if it is closed in every Hausdorff space containing it.
; [[Hemicompact space|Hemicompact]]: A space is hemicompact, if there is a sequence of compact subsets so that every compact subset is contained in one of them.
; Hereditarily ''P'': A space is hereditarily ''P'' for some property ''P'' if every subspace is also ''P''.
; [[Hereditary property|Hereditary]]: A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.<ref name=ss4>Steen &amp; Seebach p.4</ref> For example, second-countability is a hereditary property.
; [[Homeomorphism]]: If ''X'' and ''Y'' are spaces, a [[homeomorphism]] from ''X'' to ''Y'' is a [[bijection|bijective]] function ''f'' :&nbsp;''X''&nbsp;→&nbsp;''Y'' such that ''f'' and ''f''<sup>−1</sup> are continuous. The spaces ''X'' and ''Y'' are then said to be '''homeomorphic'''. From the standpoint of topology, homeomorphic spaces are identical.
; [[Homogeneous space|Homogeneous]]: A space ''X'' is [[Homogeneous space|homogeneous]] if, for every ''x'' and ''y'' in ''X'', there is a homeomorphism ''f'' : ''X'' &nbsp;→&nbsp; ''X'' such that ''f''(''x'') = ''y''. Intuitively, the space looks the same at every point. Every [[topological group]] is homogeneous.
; [[homotopic|Homotopic maps]]: Two continuous maps  ''f'', ''g'' : ''X'' &nbsp;→&nbsp; ''Y'' are [[homotopic]] (in ''Y'') if there is a continuous map ''H'' : ''X'' × [0, 1] &nbsp;→&nbsp; ''Y'' such that ''H''(''x'', 0) = ''f''(''x'') and ''H''(''x'', 1) = ''g''(''x'') for all ''x'' in ''X''. Here, ''X'' × [0, 1] is given the product topology. The function ''H'' is called a '''homotopy''' (in ''Y'') between ''f'' and ''g''.
; Homotopy: See '''[[homotopic|Homotopic maps]]'''.
; [[Hyperconnected space|Hyperconnected]]: A space is hyperconnected if no two non-empty open sets are disjoint<ref name=ss29/>  Every hyperconnected space is connected.<ref name=ss29/>

== I ==

; Identification map: See '''[[Quotient map (topology)|Quotient map]]'''.
; [[Quotient space (topology)|Identification space]]: See '''[[Quotient space (topology)|Quotient space]]'''.
; [[Indiscrete space]]: See '''[[Trivial topology]]'''.
; [[Infinite-dimensional topology]]: See '''[[Hilbert manifold]]''' and '''[[Q-manifolds]]''', i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively.
; Inner limiting set: A ''G''<sub>δ</sub> set.<ref name=ss162/>
; [[Interior (topology)|Interior]]: The [[interior (topology)|interior]] of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set ''S'' is an '''interior point''' of ''S''.
; Interior point: See '''[[Interior (topology)|Interior]]'''.
; [[Isolated point]]: A point ''x'' is an [[isolated point]] if the [[singleton (mathematics)|singleton]] {''x''} is open. More generally, if ''S'' is a subset of a space ''X'', and if ''x'' is a point of ''S'', then ''x'' is an isolated point of ''S'' if {''x''} is open in the subspace topology on ''S''.
; Isometric isomorphism: If ''M''<sub>1</sub> and ''M''<sub>2</sub> are metric spaces, an isometric isomorphism from ''M''<sub>1</sub> to ''M''<sub>2</sub> is a [[bijection|bijective]] isometry ''f'' : ''M''<sub>1</sub> &nbsp;→&nbsp; ''M''<sub>2</sub>. The metric spaces are then said to be '''isometrically isomorphic'''. From the standpoint of metric space theory, isometrically isomorphic spaces are identical.
; Isometry: If (''M''<sub>1</sub>, ''d''<sub>1</sub>) and (''M''<sub>2</sub>, ''d''<sub>2</sub>) are metric spaces, an isometry from ''M''<sub>1</sub> to ''M''<sub>2</sub> is a function ''f'' : ''M''<sub>1</sub> &nbsp;→&nbsp; ''M''<sub>2</sub> such that ''d''<sub>2</sub>(''f''(''x''), ''f''(''y'')) = ''d''<sub>1</sub>(''x'', ''y'') for all ''x'', ''y'' in ''M''<sub>1</sub>. Every isometry is [[Injective function|injective]], although not every isometry is [[surjection|surjective]].

== K ==

;[[Kolmogorov space|Kolmogorov axiom]]: See '''[[T0 space|T<sub>0</sub>]]'''.

;[[Kuratowski closure axioms]]: The [[Kuratowski closure axioms]] is a set of [[axiom]]s satisfied by the function which takes each subset of ''X'' to its closure:
:# ''[[Isotone function|Isotonicity]]'': Every set is contained in its closure.
:# ''[[Idempotent function|Idempotence]]'': The closure of the closure of a set is equal to the closure of that set.
:# ''Preservation of binary unions'': The closure of the union of two sets is the union of their closures.
:# ''Preservation of nullary unions'': The closure of the empty set is empty.
:If ''c'' is a function from the [[power set]] of ''X'' to itself, then ''c'' is a '''closure operator''' if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on ''X'' by declaring the closed sets to be the [[fixed point (mathematics)|fixed point]]s of this operator, i.e. a set ''A'' is closed [[if and only if]] ''c''(''A'') = ''A''.

;Kolmogorov topology
:T<sub>''Kol''</sub> = {R, <math>\varnothing</math>}∪{(a,∞): a is real number}; the pair (R,T<sub>''Kol''</sub>) is named ''Kolmogorov Straight''.

== L ==

;L-space: An ''L-space'' is a [[Hereditary property#In topology|hereditarily]] [[Lindelöf space]] which is not hereditarily [[Separable space|separable]].  A [[Suslin line]] would be an L-space.<ref name=GKW290>{{cite book | title=Sets and Extensions in the Twentieth Century | editor1-first=Dov M. | editor1-last=Gabbay | editor2-first=Akihiro | editor2-last=Kanamori | editor3-first=John Hayden | editor3-last=Woods | publisher=Elsevier | year=2012 | isbn=978-0444516213 | page=290 }}</ref>

;Larger topology: See '''[[Finer topology]]'''.

;[[Limit point]]: A point ''x'' in a space ''X'' is a [[limit point]] of a subset ''S'' if every open set containing ''x'' also contains a point of ''S'' other than ''x'' itself. This is equivalent to requiring that every neighbourhood of ''x'' contains a point of ''S'' other than ''x'' itself.

;Limit point compact: See '''Weakly countably compact'''.

;[[Lindelöf space|Lindelöf]]: A space is [[Lindelöf space|Lindelöf]] if every open cover has a [[countable]] subcover.

;[[Local base]]: A set ''B'' of neighbourhoods of a point ''x'' of a space ''X'' is a local base (or '''local basis''', '''neighbourhood base''', '''neighbourhood basis''') at ''x'' if every neighbourhood of ''x'' contains some member of ''B''.

;Local basis: See '''Local base'''.

;Locally (P) space: There are two definitions for a space to be "locally (P)" where (P) is a topological or set-theoretic property: that each point has a neighbourhood with property (P), or that every point has a neighourbood base for which each member has property (P).  The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected.<ref name=EGT65>Hart et al (2004) p.65</ref>

;[[Locally closed subset]]: A subset of a topological space that is the intersection of an open and a closed subset.  Equivalently, it is a relatively open subset of its closure.

;[[Locally compact space|Locally compact]]: A space is [[Locally compact space|locally compact]] if every point has a compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces.<ref name=EGT65/>  Every locally compact Hausdorff space is Tychonoff.

;[[Locally connected]]: A space is [[locally connected]] if every point has a local base consisting of connected neighbourhoods.<ref name=EGT65/>

; Locally dense: see ''Preopen''.

; [[Locally finite collection|Locally finite]]: A collection of subsets of a space is [[Locally finite collection|locally finite]] if every point has a neighbourhood which has nonempty intersection with only [[finite set|finite]]ly many of the subsets.  See also '''countably locally finite''', '''[[point finite]]'''.

;Locally metrizable'''/'''Locally metrisable: A space is locally metrizable if every point has a metrizable neighbourhood.<ref name=EGT65/>

;[[Locally path-connected]]: A space is [[locally path-connected]] if every point has a local base consisting of path-connected neighbourhoods.<ref name=EGT65/>  A locally path-connected space is connected [[if and only if]] it is path-connected.

;[[Locally simply connected]]: A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods.

;[[Loop (topology)|Loop]]: If ''x'' is a point in a space ''X'', a [[loop (topology)|loop]] at ''x'' in ''X'' (or a loop in ''X'' with basepoint ''x'') is a path ''f'' in ''X'', such that ''f''(0) = ''f''(1) = ''x''. Equivalently, a loop in ''X'' is a continuous map from the [[unit circle]] ''S''<sup>1</sup> into ''X''.

== M ==

;[[Meagre set|Meagre]]: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is meagre in ''X'' (or of '''first category''' in ''X'') if it is the [[countable]] union of nowhere dense sets.  If ''A'' is not meagre in ''X'', ''A'' is of '''second category''' in ''X''.<ref name=ss7>Steen &amp; Seebach (1978) p.7</ref>

;[[Metacompact space|Metacompact]]: A space is metacompact if every open cover has a point finite open refinement.

;Metric: See '''[[Metric space]]'''.

;Metric invariant: A metric invariant is a property which is preserved under isometric isomorphism.

;[[Metric map]]: If ''X'' and ''Y'' are metric spaces with metrics ''d''<sub>''X''</sub> and ''d''<sub>''Y''</sub> respectively, then a [[metric map]] is a function ''f'' from ''X'' to ''Y'', such that for any points ''x'' and ''y'' in ''X'', ''d''<sub>''Y''</sub>(''f''(''x''), ''f''(''y'')) ≤ ''d''<sub>''X''</sub>(''x'', ''y''). A metric map is [[metric map|strictly metric]] if the above inequality is strict for all ''x'' and ''y'' in ''X''.

;[[Metric space]]: A [[metric space]] (''M'', ''d'') is a set ''M'' equipped with a function ''d'' :&nbsp;''M''&nbsp;×&nbsp;''M''&nbsp;→&nbsp;'''[[real number|R]]''' satisfying the following axioms for all ''x'', ''y'', and ''z'' in ''M'':
:# ''d''(''x'', ''y'') ≥ 0
:# ''d''(''x'', ''x'') = 0
:# if &nbsp; ''d''(''x'', ''y'') = 0 &nbsp; then &nbsp; ''x'' = ''y'' &nbsp;&nbsp;&nbsp; (''identity of indiscernibles'')
:# ''d''(''x'', ''y'') = ''d''(''y'', ''x'')  &nbsp;&nbsp;&nbsp; (''symmetry'')
:# ''d''(''x'', ''z'') ≤ ''d''(''x'', ''y'') + ''d''(''y'', ''z'') &nbsp;&nbsp;&nbsp;  (''[[triangle inequality]]'')

:The function ''d'' is a '''metric''' on ''M'', and ''d''(''x'', ''y'') is the '''distance''' between ''x'' and ''y''. The collection of all open balls of ''M'' is a base for a topology on ''M''; this is the topology on ''M'' induced by ''d''. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable.

;[[Metrizable]]'''/'''Metrisable: A space is [[metrizable]] if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable.

;Monolith: Every non-empty ultra-connected compact space ''X'' has a largest proper open subset; this subset is called a '''monolith'''.

;[[Moore space (topology)|Moore space]]: A [[Moore space (topology)|Moore space]] is a [[developable space|developable]] [[regular Hausdorff space]].<ref name=ss163>Steen &amp; Seebach (1978) p.163</ref>

== N ==

; Nearly open: see ''preopen''. See also: [[almost open map]].

;[[Neighbourhood (mathematics)|Neighbourhood]]'''/'''Neighborhood: A neighbourhood of a point ''x'' is a set containing an open set which in turn contains the point ''x''. More generally, a neighbourhood of a set ''S'' is a set containing an open set which in turn contains the set ''S''. A neighbourhood of a point ''x'' is thus a neighbourhood of the [[singleton (mathematics)|singleton]] set {''x''}. (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.)

;[[Local base|Neighbourhood base]]/basis: See '''[[Local base]]'''.

;Neighbourhood system for a point ''x'': A [[neighbourhood system]] at a point ''x'' in a space is the collection of all neighbourhoods of ''x''.

;[[Net (mathematics)|Net]]: A [[net (mathematics)|net]] in a space ''X'' is a map from a [[directed set]] ''A'' to ''X''. A net from ''A'' to ''X'' is usually denoted (''x''<sub>α</sub>), where α is an [[index set|index variable]] ranging over ''A''. Every [[sequence]] is a net, taking ''A'' to be the directed set of [[natural number]]s with the usual ordering.

;[[Normal space|Normal]]: A space is [[normal space|normal]] if any two disjoint closed sets have disjoint neighbourhoods.<ref name=ss162/>  Every normal space admits a [[partition of unity]].

;[[T4 space|Normal Hausdorff]]: A [[T4 space|normal Hausdorff]] space (or [[T4 space|'''T<sub>4</sub>''' space]]) is a normal T<sub>1</sub> space. (A normal space is Hausdorff [[if and only if]] it is T<sub>1</sub>, so the terminology is consistent.) Every normal Hausdorff space is Tychonoff.

;[[Nowhere dense set|Nowhere dense]]: A [[nowhere dense set]] is a set whose closure has empty interior.

== O ==

; [[Open cover]]: An [[open cover]] is a cover consisting of open sets.<ref name=ss163/>

; Open ball: If (''M'', ''d'') is a metric space, an open ball is a set of the form ''B''(''x''; ''r'') := {''y'' in ''M'' : ''d''(''x'', ''y'') < ''r''}, where ''x'' is in ''M'' and ''r'' is a [[positive number|positive]] [[real number]], the '''radius''' of the ball. An open ball of radius ''r'' is an '''open ''r''-ball'''. Every open ball is an open set in the topology on ''M'' induced by ''d''.
; Open condition:  See '''open property'''.
; [[Open set]]: An [[open set]] is a member of the topology.
; [[Open map|Open function]]: A function from one space to another is [[open map|open]] if the [[image (mathematics)|image]] of every open set is open.
; Open property:  A property of points in a [[topological space]] is said to be "open" if those points which possess it form an [[open set]].  Such conditions often take a common form, and that form can be said to be an ''open condition''; for example, in [[metric space]]s, one defines an open ball as above, and says that "strict inequality is an open condition".

;[[Orthocompact space|Orthocompact]]
: A space is orthocompact, if every [[open cover]] has an interior-preserving open [[Refinement (topology)|refinement]].

== P ==<!-- This section is linked from [[Closure (topology)]] -->

;[[Paracompact space|Paracompact]]: A space is [[paracompact space|paracompact]] if every open cover has a locally finite open refinement.  Paracompact implies metacompact.<ref name=ss23>Steen &amp; Seebach (1978) p.23</ref>  Paracompact Hausdorff spaces are normal.<ref name=ss25>Steen &amp; Seebach (1978) p.25</ref>

;[[Partition of unity]]: A partition of unity of a space ''X'' is a set of continuous functions from ''X'' to [0, 1] such that any point has a neighbourhood where all but a [[finite set|finite]] number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.

;[[Path (topology)|Path]]: A [[Path (topology)|path]] in a space ''X'' is a continuous map ''f'' from the closed unit [[interval (mathematics)|interval]] [0, 1] into ''X''. The point ''f''(0) is the initial point of ''f''; the point ''f''(1) is the terminal point of ''f''.<ref name=ss29>Steen &amp; Seebach (1978) p.29</ref>

;[[Path-connected space|Path-connected]]: A space ''X'' is [[path-connected space|path-connected]] if, for every two points ''x'', ''y'' in ''X'', there is a path ''f'' from ''x'' to ''y'', i.e., a path with initial point ''f''(0) = ''x'' and terminal point ''f''(1) = ''y''. Every path-connected space is connected.<ref name=ss29/>

;Path-connected component: A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a [[partition of a set|partition]] of that space, which is [[partition of a set|finer]] than the partition into connected components.<ref name=ss29/> The set of path-connected components of a space ''X'' is denoted [[homotopy groups|π<sub>0</sub>(''X'')]].

;Perfectly normal: a normal space which is also a G<sub>δ</sub>.<ref name=ss162/>

;π-base: A collection ''B'' of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from ''B''.<ref>Hart, Nagata, Vaughan Sect. d-22, page 227</ref>

;Point: A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point".

;Point of closure: See '''[[Closure (topology)|Closure]]'''.

;[[Polish space|Polish]]: A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space.

;[[Polyadic space|Polyadic]]: A space is polyadic if it is the continuous image of the power of a [[one-point compactification]] of a locally compact, non-compact Hausdorff space.

;[[Polytopological space]]: A polytopological space is a [[set (mathematics)|set]] <math>X</math> together with a [[family of sets|family]] <math>\{\tau_i\}_{i\in I}</math> of [[Topological_space#Definition_via_open_sets|topologies]] on <math>X</math> that is [[Total order|linearly ordered]] by the [[Subset|inclusion relation]] where <math>I</math> is an arbitrary [[index set]].

;P-point: A point of a topological space is a P-point if its [[Neighbourhood system|filter of neighbourhoods]] is closed under countable intersections.

;Pre-compact: See '''[[Relatively compact]]'''.

;{{visible anchor|Pre-open set|Preopen set}}: A subset ''A'' of a topological space ''X'' is preopen if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X A \right)</math>.{{sfn|Hart|Nagata|Vaughan|2004|pp=8–9}}

;Prodiscrete topology: The prodiscrete topology on a product ''A''<sup>''G''</sup> is the product topology when each factor ''A'' is given the discrete topology.<ref>{{cite book | last1=Ceccherini-Silberstein | first1=Tullio | last2=Coornaert | first2=Michel | title=Cellular automata and groups | zbl=1218.37004 | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | isbn=978-3-642-14033-4 | year=2010 | page=3 }}</ref>

;[[Product topology]]: If <math>\left(X_i\right)</math> is a collection of spaces and ''X'' is the (set-theoretic) [[Cartesian product]] of <math>\left(X_i\right),</math> then the [[product topology]] on ''X'' is the coarsest topology for which all the projection maps are continuous.

;Proper function/mapping: A continuous function ''f'' from a space ''X'' to a space ''Y'' is proper if <math>f^{-1}(C)</math> is a compact set in ''X'' for any compact subspace ''C'' of ''Y''.

;[[Proximity space]]: A proximity space (''X'',&nbsp;'''d''') is a set ''X'' equipped with a [[binary relation]] '''d''' between subsets of ''X'' satisfying the following properties:

:For all subsets ''A'', ''B'' and ''C'' of ''X'',
:#''A'' '''d''' ''B'' implies ''B'' '''d''' ''A''
:#''A'' '''d''' ''B'' implies ''A'' is non-empty
:#If ''A'' and ''B'' have non-empty intersection, then ''A'' '''d''' ''B''
:#''A'' '''d''' (''B''&nbsp;<math>\cup</math>&nbsp;''C'') [[if and only if]] (''A'' '''d''' ''B'' or ''A'' '''d''' ''C'')
:#If, for all subsets ''E'' of ''X'', we have (''A'' '''d''' ''E'' or ''B'' '''d''' ''E''), then we must have ''A'' '''d''' (''X'' − ''B'')

;[[Pseudocompact]]: A space is pseudocompact if every [[real number|real-valued]] continuous function on the space is bounded.

;Pseudometric: See '''Pseudometric space'''.

;[[Pseudometric space]]: A pseudometric space (''M'', ''d'') is a set ''M'' equipped with a [[Real number|real]]-valued function <math>d : M \times M \to \R</math> satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical.  The function ''d'' is a '''pseudometric''' on ''M''. Every metric is a pseudometric.

;Punctured neighbourhood'''/'''Punctured neighborhood: A punctured neighbourhood of a point ''x'' is a neighbourhood of ''x'', [[Set subtraction|minus]] {''x''}. For instance, the [[interval (mathematics)|interval]] (−1, 1) = {''y'' : −1 < ''y'' < 1} is a neighbourhood of ''x'' = 0 in the [[real line]], so the set <math>(-1, 0) \cup (0, 1) = (-1, 1) - \{ 0 \}</math> is a punctured neighbourhood of 0.

== Q ==

;Quasicompact: See '''[[compact space|compact]]'''. Some authors define "compact" to include the [[Hausdorff space|Hausdorff]] separation axiom, and they use the term '''quasicompact''' to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French.

;[[Quotient map (topology)|Quotient map]]: If ''X'' and ''Y'' are spaces, and if ''f'' is a [[surjection]] from ''X'' to ''Y'', then ''f'' is a quotient map (or '''identification map''') if, for every subset ''U'' of ''Y'', ''U'' is open in ''Y'' [[if and only if]] ''f''<sup>&nbsp;{{mono|-}}1</sup>(''U'') is open in ''X''.  In other words, ''Y'' has the ''f''-strong topology.  Equivalently, <math>f</math> is a quotient map if and only if it is the transfinite composition of maps <math>X\rightarrow X/Z</math>, where <math>Z\subset X</math> is a subset. Note that this does not imply that ''f'' is an open function.

;[[Quotient space (topology)|Quotient space]]: If ''X'' is a space, ''Y'' is a set, and ''f'' :&nbsp;''X''&nbsp;→&nbsp;''Y'' is any [[surjection|surjective]] function, then the [[Quotient topology]] on ''Y'' induced by ''f'' is the finest topology for which ''f'' is continuous. The space ''X'' is a quotient space or '''identification space'''. By definition, ''f'' is a quotient map. The most common example of this is to consider an [[equivalence relation]] on ''X'', with ''Y'' the set of [[equivalence class]]es and ''f'' the natural projection map. This construction is dual to the construction of the subspace topology.

== R ==

; Refinement: A cover ''K'' is a [[refinement (topology)|refinement]] of a cover ''L'' if every member of ''K'' is a subset of some member of ''L''.
; [[Regular space|Regular]]: A space is [[regular space|regular]] if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and ''x'' have [[Disjoint sets|disjoint]] neighbourhoods.
; [[T3 space|Regular Hausdorff]]: A space is [[T3 space|regular Hausdorff]] (or '''T<sub>3</sub>''') if it is a regular T<sub>0</sub> space.  (A regular space is Hausdorff [[if and only if]] it is T<sub>0</sub>, so the terminology is consistent.)
; {{visible anchor|[[Regular open set|Regular open]]}}: A subset of a space ''X'' is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior.<ref name="Steen & Seebach 1978 p.6">Steen & Seebach (1978) p.6</ref>  An example of a non-regular open set is the set  ''U'' = {{open-open|0,1}} ∪ {{open-open|1,2}} in '''R''' with its normal topology, since 1 is in the interior of the closure of ''U'', but not in ''U''. The regular open subsets of a space form a [[complete Boolean algebra]].<ref name="Steen & Seebach 1978 p.6"/>
; [[Relatively compact]]: A subset ''Y'' of a space ''X'' is [[relatively compact]] in ''X'' if the closure of ''Y'' in ''X'' is compact.
; Residual: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is residual in ''X'' if the complement of ''A'' is meagre in ''X''. Also called '''comeagre''' or '''comeager'''.
; Resolvable: A [[topological space]] is called [[resolvable space|resolvable]] if it is expressible as the union of two [[disjoint sets|disjoint]] [[dense subset]]s.
; Rim-compact: A space is rim-compact if it has a base of open sets whose boundaries are compact.

== S ==

;S-space:  An ''S-space'' is a [[Hereditary property#In topology|hereditarily]] [[separable space]] which is not hereditarily [[Lindelöf space|Lindelöf]].<ref name=GKW290/>

;[[Scattered space|Scattered]]: A space ''X'' is [[scattered space|scattered]] if every nonempty subset ''A'' of ''X'' contains a point isolated in ''A''.

;[[Scott continuity|Scott]]: The [[Scott topology]] on a [[poset]] is that in which the open sets are those [[Upper set]]s inaccessible by directed joins.<ref>Vickers (1989) p.95</ref>

;Second category: See '''Meagre'''.

;[[Second-countable space|Second-countable]]: A space is [[second-countable space|second-countable]] or '''perfectly separable''' if it has a [[countable]] base for its topology.<ref name=ss162/> Every second-countable space is first-countable, separable, and Lindelöf.

;[[Semilocally simply connected]]: A space ''X'' is [[semilocally simply connected]] if, for every point ''x'' in ''X'', there is a neighbourhood ''U'' of ''x'' such that every loop at ''x'' in ''U'' is homotopic in ''X'' to the constant loop ''x''. Every simply connected space and every locally simply connected space is semilocally simply connected. (Compare with locally simply connected; here, the homotopy is allowed to live in ''X'', whereas in the definition of locally simply connected, the homotopy must live in ''U''.)

;Semi-open: A subset ''A'' of a topological space ''X'' is called semi-open if <math>A \subseteq \operatorname{Cl}_X \left( \operatorname{Int}_X A \right)</math>.{{sfn|Hart|Nagata|Vaughan|2004|p=8}}

;Semi-preopen: A subset ''A'' of a topological space ''X'' is called semi-preopen if <math>A \subseteq \operatorname{Cl}_X \left( \operatorname{Int}_X \left( \operatorname{Cl}_X A \right) \right)</math>{{sfn|Hart|Nagata|Vaughan|2004|p=9}}

;[[semiregular space|Semiregular]]: A space is semiregular if the regular open sets form a base.

;[[Separable (topology)|Separable]]: A space is [[separable (topology)|separable]] if it has a [[countable]] dense subset.<ref name=ss162/><ref name=ss7/>

;[[Separated sets|Separated]]: Two sets ''A'' and ''B'' are [[separated sets|separated]] if each is [[Disjoint sets|disjoint]] from the other's closure.

;[[Sequentially compact]]: A space is sequentially compact if every [[sequence]] has a convergent subsequence. Every sequentially compact space is countably compact, and every first-countable, countably compact space is sequentially compact.

;[[Short map]]: See '''[[metric map]]'''

;[[Simply connected space|Simply connected]]: A space is [[simply connected space|simply connected]] if it is path-connected and every loop is homotopic to a constant map.

;Smaller topology: See '''[[Coarser topology]]'''.

;[[Sober space|Sober]]: In a [[sober space]], every [[hyperconnected space|irreducible]] closed subset is the [[closure (topology)|closure]] of exactly one point: that is, has a unique [[generic point]].<ref>Vickers (1989) p.66</ref>

;Star: The star of a point in a given [[cover (topology)|cover]] of a [[topological space]] is the union of all the sets in the cover that contain the point. See '''[[star refinement]]'''.

;<math>f</math>-Strong topology:  Let <math>f\colon X\rightarrow Y</math> be a map of topological spaces.  We say that <math>Y</math> has the <math>f</math>-strong topology if, for every subset <math>U\subset Y</math>, one has that <math>U</math> is open in <math>Y</math> if and only if <math>f^{-1}(U)</math> is open in <math>X</math>

;Stronger topology: See '''[[Finer topology]]'''. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''weaker topology'''.

;[[Subbase]]: A collection of open sets is a [[subbase]] (or '''subbasis''') for a topology if every non-empty proper open set in the topology is the union of a [[finite set|finite]] intersection of sets in the subbase. If <math>\mathcal B</math> is ''any'' collection of subsets of a set ''X'', the topology on ''X'' generated by <math>\mathcal B</math> is the smallest topology containing <math>\mathcal B;</math> this topology consists of the empty set, ''X'' and all unions of finite intersections of elements of <math>\mathcal B.</math>  Thus <math>\mathcal B</math> is a subbase for the topology it generates.

;[[Subbase|Subbasis]]: See '''[[Subbase]]'''.

;Subcover: A cover ''K'' is a subcover (or '''subcovering''') of a cover ''L'' if every member of ''K'' is a member of ''L''.

;Subcovering: See '''Subcover'''.
;[[Submaximal space]]: A [[topological space]] is said to be submaximal if every subset of it is locally closed, that is, every subset is the intersection of an [[open set]] and a [[closed set]].

Here are some facts about submaximality as a property of topological spaces:

* Every [[door space]] is submaximal.
* Every submaximal space is ''weakly submaximal'' viz every finite set is locally closed.
* Every submaximal space is [[irresolvable space|irresolvable]].<ref>{{citation | title=Recent progress in general topology | volume=2 | author1=Miroslav Hušek | author2=J. van Mill | publisher=Elsevier | year=2002 | isbn=0-444-50980-1 | page=21 }}</ref>

;Subspace: If ''T'' is a topology on a space ''X'', and if ''A'' is a subset of ''X'', then the [[subspace topology]] on ''A'' induced by ''T'' consists of all intersections of open sets in ''T'' with ''A''. This construction is dual to the construction of the quotient topology.

;[[Suslin line]]

== T ==

;[[T0 space|T<sub>0</sub>]]: A space is [[T0 space|T<sub>0</sub>]] (or '''Kolmogorov''') if for every pair of distinct points ''x'' and ''y'' in the space, either there is an open set containing ''x'' but not ''y'', or there is an open set containing ''y'' but not ''x''.

;[[T1 space|T<sub>1</sub>]]: A space is [[T1 space|T<sub>1</sub>]] (or '''Fréchet''' or '''accessible''') if for every pair of distinct points ''x'' and ''y'' in the space, there is an open set containing ''x'' but not ''y''. (Compare with T<sub>0</sub>; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T<sub>1</sub> if all its [[singleton (mathematics)|singleton]]s are closed. Every T<sub>1</sub> space is T<sub>0</sub>.

;[[T2 space|T<sub>2</sub>]]: See '''[[Hausdorff space]]'''.

;[[T3 space|T<sub>3</sub>]]: See '''[[T3 space|Regular Hausdorff]]'''.

;[[Tychonoff space|T<sub>3½</sub>]]: See '''[[Tychonoff space]]'''.

;[[T4 space|T<sub>4</sub>]]: See '''[[T4 space|Normal Hausdorff]]'''.

;[[T5 space|T<sub>5</sub>]]: See '''[[T5 space|Completely normal Hausdorff]]'''.

;[[Category of topological spaces|Top]]: See '''[[Category of topological spaces]]'''.

;θ-cluster point, θ-closed, θ-open: A point ''x'' of a topological space ''X'' is a θ-cluster point of a subset ''A'' if <math>A \cap \operatorname{Cl}_X(U) \neq \emptyset</math> for every open neighborhood ''U'' of ''x'' in ''X''.  The subset ''A'' is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed.{{sfn|Hart|Nagata|Vaughan|2004|p=8}}

;[[Topological invariant]]: A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. [[Algebraic topology]] is the study of topologically invariant [[abstract algebra]] constructions on topological spaces.

;[[Topological space]]: A [[topological space]] (''X'', ''T'') is a set ''X'' equipped with a collection ''T'' of subsets of ''X'' satisfying the following [[axiom]]s:

:# The empty set and ''X'' are in ''T''.
:# The union of any collection of sets in ''T'' is also in ''T''.
:# The intersection of any pair of sets in ''T'' is also in ''T''.

:The collection ''T'' is a '''topology''' on ''X''.

;Topological sum: See '''Coproduct topology'''.

;Topologically complete: [[Completely metrizable space]]s (i. e. topological spaces homeomorphic to complete metric spaces) are often called ''topologically complete''; sometimes the term is also used for [[Čech-complete space]]s or [[completely uniformizable space]]s.

;Topology: See '''[[Topological space]]'''.

;Totally bounded: A metric space ''M'' is totally bounded if, for every ''r'' > 0, there exist a [[finite set|finite]] cover of ''M'' by open balls of radius ''r''. A metric space is compact if and only if it is complete and totally bounded.

;Totally disconnected: A space is totally disconnected if it has no connected subset with more than one point.

;[[Trivial topology]]: The [[trivial topology]] (or '''indiscrete topology''') on a set ''X'' consists of precisely the empty set and the entire space ''X''.

;[[Tychonoff space|Tychonoff]]: A [[Tychonoff space]] (or '''completely regular Hausdorff''' space, '''completely T<sub>3</sub>''' space, '''T<sub>3.5</sub>''' space) is a completely regular T<sub>0</sub> space.  (A completely regular space is Hausdorff [[if and only if]] it is T<sub>0</sub>, so the terminology is consistent.) Every Tychonoff space is regular Hausdorff.

== U ==

;Ultra-connected: A space is ultra-connected if no two non-empty closed sets are disjoint.<ref name="ss29"/> Every ultra-connected space is path-connected.

;[[Ultrametric space|Ultrametric]]: A metric is an ultrametric if it satisfies the following stronger version of the [[triangle inequality]]: for all ''x'', ''y'', ''z'' in ''M'', ''d''(''x'', ''z'') ≤ max(''d''(''x'', ''y''), ''d''(''y'', ''z'')).

;[[Uniform isomorphism]]: If ''X'' and ''Y'' are [[uniform space]]s, a uniform isomorphism from ''X'' to ''Y'' is a bijective function ''f'' : ''X'' → ''Y'' such that ''f'' and ''f''<sup>−1</sup> are [[uniformly continuous]]. The spaces are then said to be uniformly isomorphic and share the same [[uniform properties]].

;[[Uniformizable]]/Uniformisable: A space is uniformizable if it is homeomorphic to a uniform space.

;[[Uniform space]]: A [[uniform space]] is a set ''X'' equipped with a nonempty collection Φ of subsets of the [[Cartesian product]] ''X'' × ''X'' satisfying the following [[axiom]]s:

:# if ''U'' is in Φ, then ''U'' contains { (''x'', ''x'') | ''x'' in ''X'' }.
:# if ''U'' is in Φ, then { (''y'', ''x'') | (''x'', ''y'') in ''U'' } is also in Φ
:# if ''U'' is in Φ and ''V'' is a subset of ''X'' × ''X'' which contains ''U'', then ''V'' is in Φ
:# if ''U'' and ''V'' are in Φ, then ''U'' ∩ ''V'' is in Φ
:# if ''U'' is in Φ, then there exists ''V'' in Φ such that, whenever (''x'', ''y'') and (''y'', ''z'') are in ''V'', then (''x'', ''z'') is in ''U''.

:The elements of Φ are called '''entourages''', and Φ itself is called a '''uniform structure''' on ''X''. The uniform structure induces a topology on ''X'' where the basic neighborhoods of ''x'' are sets of the form {''y'' : (''x'',''y'')∈''U''} for ''U''∈Φ.

;Uniform structure: See '''[[Uniform space]]'''.

== W ==

; [[Weak topology]]: The [[weak topology]] on a set, with respect to a collection of functions from that set into topological spaces, is the coarsest topology on the set which makes all the functions continuous.
; Weaker topology: See '''[[Coarser topology]]'''. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''stronger topology'''.
; Weakly countably compact: A space is weakly countably compact (or '''limit point compact''') if every [[Infinity|infinite]] subset has a limit point.
; Weakly hereditary: A property of spaces is said to be weakly hereditary if whenever a space has that property, then so does every closed subspace of it. For example, compactness and the Lindelöf property are both weakly hereditary properties, although neither is hereditary.
; Weight: The [[weight of a space]] ''X'' is the smallest [[cardinal number]] κ such that ''X'' has a base of cardinal κ. (Note that such a cardinal number exists, because the entire topology forms a base, and because the class of cardinal numbers is [[well order|well-ordered]].)
; Well-connected: See '''Ultra-connected'''. (Some authors use this term strictly for ultra-connected compact spaces.)

== Z ==

;Zero-dimensional: A space is [[zero-dimensional]] if it has a base of clopen sets.<ref name=ss33>Steen &amp; Seebach (1978) p.33</ref>

== See also ==
* [[Naive set theory]], [[Axiomatic set theory]], and [[Function (mathematics)|Function]] for definitions concerning sets and functions. 
* [[Topology]] for a brief history and description of the subject area
* [[Topological space]]s for basic definitions and examples
* [[List of general topology topics]]
* [[List of examples in general topology]]
;Topology specific concepts
* [[Compact space]]
* [[Connected space]]
* [[Continuity (topology)|Continuity]]
* [[Metric space]]
* [[Separated sets]]
* [[Separation axiom]]
* [[Topological space]]
* [[Uniform space]]
;Other glossaries
* [[Glossary of algebraic topology]]
*[[Glossary of differential geometry and topology]]
* [[Glossary of areas of mathematics]]
* [[Glossary of Riemannian and metric geometry]]

== References ==
{{reflist|30em}}
* {{cite book | title=Encyclopedia of general topology | first1=Klaas Pieter | last1=Hart | first2=Jun-iti | last2=Nagata | first3=Jerry E. | last3=Vaughan | publisher=Elsevier | year=2004 | isbn=978-0-444-50355-8 }}
*{{cite book|title=Handbook of Set-Theoretic Topology |year=1984 |editor-last=Kunen |editor-first=Kenneth |editor-link=Kenneth Kunen |editor2=Vaughan, Jerry E. |publisher=North-Holland|isbn=0-444-86580-2}}
* {{cite book | last=Nagata | first=Jun-iti | author-link=Jun-iti Nagata | title=Modern general topology | edition=2nd revised | zbl=0598.54001 | series=North-Holland Mathematical Library | volume=33 | location=Amsterdam-New York-Oxford | publisher=North-Holland | year=1985 | isbn=0080933793 }}
*{{cite book | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 }}
*{{cite book | first=Steven | last=Vickers | author-link=Steve Vickers (computer scientist) | title=Topology via Logic | series=Cambridge Tracts in Theoretic Computer Science | volume=5 | isbn=0-521-36062-5 | year=1989 | zbl=0668.54001 }}
*{{cite book | first=Stephen | last=Willard | title=General Topology | year=1970 | publisher=Addison-Wesley | location=Reading, MA | zbl=0205.26601 | series=Addison-Wesley Series in Mathematics | isbn=978-0-201-08707-9 | url-access=registration | url=https://archive.org/details/generaltopology00will_0 }} Also available as Dover reprint.

== External links ==
* [https://web.archive.org/web/20061205203826/http://math.berkeley.edu/~apollo/topodefs.ps A glossary of definitions in topology]

{{Topology}}

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