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{{Short description|Property of topological spaces}}
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[[Image:Neighborhood illust1.svg|right|thumb|In this topological space, ''V'' is a neighbourhood of ''p'' and it contains a connected open set (the dark green disk) that contains ''p''.]]

In [[topology]] and other branches of [[mathematics]], a [[topological space]] ''X'' is
'''locally connected''' if every point admits a [[neighbourhood basis]] consisting of [[Open set|open]] [[Connected set|connected]] sets.

As a stronger notion, the space ''X'' is '''locally path connected''' if every point admits a neighbourhood basis consisting of open [[path connected]] sets.

==Background==
Throughout the history of topology, [[Connected space|connectedness]] and [[Compact space|compactness]] have been two of the most
widely studied topological properties.  Indeed, the study of these properties even among subsets of [[Euclidean space]], and the recognition of their independence from the particular form of the [[Euclidean metric]], played a large role in clarifying the notion of a topological property and thus a topological space.  However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the [[Heine–Borel theorem]], ''connected'' subsets of <math>\R^n</math> (for ''n'' > 1) proved to be much more complicated.  Indeed, while any compact [[Hausdorff space]] is [[locally compact]], a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below).

This led to a rich vein of research in the first half of the twentieth century, in which topologists studied the implications between increasingly subtle and complex variations on the notion of a locally connected space.  As an example, the notion of local connectedness im kleinen at a point and its relation to local connectedness will be considered later on in the article.

In the latter part of the twentieth century, research trends shifted to more intense study of spaces like [[manifold]]s, which are locally well understood (being [[locally homeomorphic]] to Euclidean space) but have complicated global behavior. By this it is meant that although the basic [[point-set topology]] of manifolds is relatively simple (as manifolds are essentially [[metrizable]] according to most definitions of the concept), their [[algebraic topology]] is far more complex. From this modern perspective, the stronger property of local path connectedness turns out to be more important: for instance, in order for a space to admit a [[universal cover]] it must be connected and locally path connected.

A space is locally connected if and only if for every open set {{math|U}}, the connected components of {{math|U}} (in the [[subspace topology]]) are open.  It follows, for instance, that a continuous function from a locally connected space to a [[totally disconnected]] space must be locally constant.  In fact the openness of components is so natural that one must be sure to keep in mind that it is not true in general: for instance [[Cantor space]] is totally disconnected but not [[discrete space|discrete]].

==Definitions==

Let <math>X</math> be a topological space, and let <math>x</math> be a point of <math>X.</math>

A space <math>X</math> is called '''locally connected at <math>x</math>'''{{sfn|Munkres|2000|p=161}} if every [[neighborhood (mathematics)|neighborhood]] of <math>x</math> contains a [[connected (topology)|connected]] ''open'' neighborhood of <math>x</math>, that is, if the point <math>x</math> has a [[neighborhood base]] consisting of connected open sets.  A '''locally connected space'''{{sfn|Willard|2004|loc=Definition 27.7|p=199}}{{sfn|Munkres|2000|p=161}} is a space that is locally connected at each of its points.

Local connectedness does not imply connectedness (consider two disjoint open intervals in <math>\R</math> for example); and connectedness does not imply local connectedness (see the [[topologist's sine curve]]).

{{anchor|locally path connected}}A space <math>X</math> is called '''locally path connected at <math>x</math>'''{{sfn|Munkres|2000|p=161}} if every neighborhood of <math>x</math> contains a [[path connected]] ''open'' neighborhood of <math>x</math>, that is, if the point <math>x</math> has a neighborhood base consisting of path connected open sets.  A '''locally path connected space'''{{sfn|Willard|2004|loc=Definition 27.4|p=199}}{{sfn|Munkres|2000|p=161}} is a space that is locally path connected at each of its points.

Locally path connected spaces are locally connected.  The converse does not hold (see the [[lexicographic order topology on the unit square]]).

===Connectedness im kleinen===

A space <math>X</math> is called '''connected im kleinen at <math>x</math>'''{{sfn|Willard|2004|loc=Definition 27.14|p=201}}<ref name="BBS"/> or '''weakly locally connected at <math>x</math>'''{{sfn|Munkres|2000|loc=Exercise 6|p=162}} if every neighborhood of <math>x</math> contains a connected (not necessarily open) neighborhood of <math>x</math>, that is, if the point <math>x</math> has a neighborhood base consisting of connected sets.  A space is called '''weakly locally connected''' if it is weakly locally connected at each of its points; as indicated below, this concept is in fact the same as being locally connected.

A space that is locally connected at <math>x</math> is connected im kleinen at <math>x.</math>  The converse does not hold, as shown for example by a certain infinite union of decreasing [[broom space]]s, that is connected im kleinen at a particular point, but not locally connected at that point.{{sfn|Steen|Seebach|1995|loc=Example 119.4|p=139}}{{sfn|Munkres|2000|loc=Exercise 7|p=162}}<ref>{{citation |title=Show that X is not locally connected at p |url=https://math.stackexchange.com/q/2439096 |website=Math StackExchange}}</ref>  However, if a space is connected im kleinen at each of its points, it is locally connected.{{sfn|Willard|2004|loc=Theorem 27.16|p=201}}

A space <math>X</math> is said to be '''path connected im kleinen at <math>x</math>'''<ref name="BBS">{{citation |last1=Björn |first1=Anders |last2=Björn |first2=Jana |last3=Shanmugalingam |first3=Nageswari |title=The Mazurkiewicz distance and sets that are finitely connected at the boundary |journal=Journal of Geometric Analysis |volume=26 |year=2016 |issue=2 |pages=873–897 |doi=10.1007/s12220-015-9575-9 |arxiv=1311.5122|s2cid=255549682 }}, section 2</ref> if every neighborhood of <math>x</math> contains a path connected (not necessarily open) neighborhood of <math>x</math>, that is, if the point <math>x</math> has a neighborhood base consisting of path connected sets.

A space that is locally path connected at <math>x</math> is path connected im kleinen at <math>x.</math>  The converse does not hold, as shown by the same infinite union of decreasing broom spaces as above.  However, if a space is path connected im kleinen at each of its points, it is locally path connected.<ref>{{citation |title=Definition of locally pathwise connected |url=https://math.stackexchange.com/q/2999685 |website=Math StackExchange}}</ref>{{Better source needed|date=December 2024}}

==First examples==

# For any positive integer ''n'', the Euclidean space <math>\R^n</math> is locally path connected, thus locally connected; it is also connected.
# More generally, every [[locally convex topological vector space]] is locally connected, since each point has a local base of [[Convex set|convex]] (and hence connected) neighborhoods.
# The subspace <math>S = [0,1] \cup [2,3]</math> of the real line <math>\R^1</math> is locally path connected but not connected.
# The [[topologist's sine curve]] is a subspace of the Euclidean plane that is connected, but not locally connected.{{sfn|Steen|Seebach|1995|pp=137–138}}
# The space <math>\Q</math> of [[rational numbers]] endowed with the standard Euclidean topology, is neither connected nor locally connected. 
# The [[comb space]] is path connected but not locally path connected, and not even locally connected.
# A countably infinite set endowed with the [[cofinite topology]] is locally connected (indeed, [[hyperconnected]]) but not locally path connected.{{sfn|Steen |Seebach|1995| pp=49–50}}
# The [[lexicographic order topology on the unit square]] is connected and locally connected, but not path connected, nor locally path connected.{{sfn|Steen| Seebach|1995|loc=Example 48| p= 73}}
# The [[Kirch space]] is connected and locally connected, but not path connected, and not path connected im kleinen at any point.  It is in fact [[totally path disconnected]].

A [[first-countable]] [[Hausdorff space]] <math>(X, \tau)</math> is locally path-connected if and only if <math>\tau</math> is equal to the [[final topology]] on <math>X</math> induced by the set <math>C([0, 1]; X)</math> of all continuous paths <math>[0, 1] \to (X, \tau).</math>

==Properties==

{{math theorem|A space is locally connected if and only if it is weakly locally connected.{{sfn|Willard|2004|loc=Theorem 27.16|p=201}}}}

{{collapse top|title=Proof|left=true}}
For the non-trivial direction, assume <math>X</math> is weakly locally connected.  To show it is locally connected, it is enough to show that the [[connected component (topology)|connected component]]s of open sets are open.

Let <math>U</math> be open in <math>X</math> and let <math>C</math> be a connected component of <math>U.</math> Let <math>x</math> be an element of <math>C.</math> Then <math>U</math> is a neighborhood of <math>x</math> so that there is a connected neighborhood <math>V</math> of <math>x</math> contained in <math>U.</math>  Since <math>V</math> is connected and contains <math>x,</math> <math>V</math> must be a subset of <math>C</math> (the connected component containing <math>x</math>). Therefore <math>x</math> is an interior point of <math>C.</math>  Since <math>x</math> was an arbitrary point of <math>C,</math> <math>C</math> is open in <math>X.</math> Therefore, <math>X</math> is locally connected.
{{collapse bottom}}

# Local connectedness is, by definition, a [[local property]] of topological spaces, i.e., a topological property ''P'' such that a space ''X'' possesses property ''P'' if and only if each point ''x'' in ''X'' admits a neighborhood base of sets that have property ''P''.  Accordingly, all the "metaproperties" held by a local property hold for local connectedness. In particular:
# A space is locally connected if and only if it admits a [[base (topology)|base]] of (open) connected subsets. 
# The [[Disjoint union (topology)|disjoint union]] <math>\coprod_i X_i</math> of a family <math>\{X_i\}</math> of spaces is locally connected if and only if each <math>X_i</math> is locally connected.  In particular, since a single point is certainly locally connected, it follows that any [[discrete space]] is locally connected.  On the other hand, a discrete space is [[totally disconnected]], so is connected only if it has at most one point.
# Conversely, a [[totally disconnected space]] is locally connected if and only if it is discrete.  This can be used to explain the aforementioned fact that the rational numbers are not locally connected.
# A nonempty product space <math>\prod_i X_i</math> is locally connected if and only if each <math>X_i</math> is locally connected and all but finitely many of the <math>X_i</math> are connected.{{sfn|Willard|2004|loc=Theorem 27.13|p=201}}
# Every [[hyperconnected space]] is locally connected, and connected.

==Components and path components==

The following result follows almost immediately from the definitions but will be quite useful:

Lemma: Let ''X'' be a space, and <math>\{Y_i\}</math> a family of subsets of ''X''.  Suppose that <math> \bigcap_i Y_i </math> is nonempty.  Then, if each <math>Y_i</math> is connected (respectively, path connected) then the union <math>\bigcup_i Y_i</math> is connected (respectively, path connected).{{sfn|Willard|2004|loc=Theorem 26.7a|p=192}}

Now consider two relations on a topological space ''X'': for <math>x,y \in X,</math> write:
:<math>x \equiv_c y</math> if there is a connected subset of ''X'' containing both ''x'' and ''y''; and
:<math> x \equiv_{pc} y </math> if there is a path connected subset of ''X'' containing both ''x'' and ''y''.

Evidently both relations are reflexive and symmetric.  Moreover, if ''x'' and ''y'' are contained in a connected (respectively, path connected) subset ''A'' and ''y'' and ''z'' are connected in a connected (respectively, path connected) subset ''B'', then the Lemma implies that <math>A \cup B</math> is a connected (respectively, path connected) subset containing ''x'', ''y'' and ''z''.  Thus each relation is an [[equivalence relation]], and defines a partition of ''X'' into [[equivalence classes]].  We consider these two partitions in turn.

For ''x'' in ''X'', the set <math>C_x</math> of all points ''y'' such that <math>y \equiv_c x</math> is called the [[connected component (topology)|connected component]] of ''x''.{{sfn|Willard|2004|loc=Definition 26.11|p=194}}  The Lemma implies that <math>C_x</math> is the unique maximal connected subset of ''X'' containing ''x''.{{sfn|Willard|2004|loc=Problem 26B|pp=195–196}} Since the closure of <math>C_x</math> is also a connected subset containing ''x'',{{sfn|Kelley|1975|loc= Theorem 20|p= 54}}{{sfn| Willard|2004|loc= Theorem 26.8|p=193}} it follows that <math>C_x</math> is closed.{{sfn|Willard|2004|loc= Theorem 26.12| p=194}}

If ''X'' has only finitely many connected components, then each component is the complement of a finite union of closed sets and therefore open.  In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e., <math>C_x = \{x\}</math> for all points ''x'') that are not discrete, like Cantor space.  However, the connected components of a locally connected space are also open, and thus are [[clopen sets]].{{sfn|Willard|2004|loc=Corollary 27.10|p=200}}  It follows that a locally connected space ''X'' is a topological disjoint union <math>\coprod C_x</math> of its distinct connected components.  Conversely, if for every open subset ''U'' of ''X'', the connected components of ''U'' are open, then ''X'' admits a base of connected sets and is therefore locally connected.{{sfn|Willard|2004|loc=Theorem 27.9| p= 200}}

Similarly ''x'' in ''X'', the set <math>PC_x</math> of all points ''y'' such that <math>y \equiv_{pc} x</math> is called the ''path component'' of ''x''.{{sfn|Willard|2004|loc=Problem 27D|p=202}} As above, <math>PC_x</math> is also the union of all path connected subsets of ''X'' that contain ''x'', so by the Lemma is itself path connected.  Because path connected sets are connected, we have <math>PC_x \subseteq C_x</math> for all <math>x \in X.</math>

However the closure of a path connected set need not be path connected: for instance, the topologist's sine curve is the closure of the open subset ''U'' consisting of all points ''(x,sin(x))'' with ''x > 0'', and ''U'', being homeomorphic to an interval on the real line, is certainly path connected.  Moreover, the path components of the topologist's sine curve ''C'' are ''U'', which is open but not closed, and <math>C \setminus U,</math> which is closed but not open.

A space is locally path connected if and only if for all open subsets ''U'', the path components of ''U'' are open.{{sfn|Willard|2004|loc=Problem 27D|p=202}} Therefore the path components of a locally path connected space give a partition of ''X'' into pairwise disjoint open sets.  It follows that an open connected subspace of a locally path connected space is necessarily path connected.{{sfn|Willard|2004|loc=Theorem 27.5|p=199}}  Moreover, if a space is locally path connected, then it is also locally connected, so for all <math>x \in X,</math> <math>C_x</math> is connected and open, hence path connected, that is, <math>C_x = PC_x.</math>  That is, for a locally path connected space the components and path components coincide.

===Examples===

# The set <math>I \times I</math>  (where <math>I = [0, 1]</math>) in the [[Lexicographical order|dictionary]] [[order topology]] has exactly one component (because it is connected) but has uncountably many path components. Indeed, any set of the form <math>\{a\} \times I</math> is a path component for each ''a'' belonging to ''I''.
# Let <math>f : \R \to \R_{\ell}</math> be a continuous map from <math>\R</math> to <math>\R_{\ell}</math> (which is <math>\R</math> in the [[lower limit topology]]). Since <math>\R</math>  is connected, and the image of a connected space under a continuous map must be connected, the image of <math>\R</math> under <math>f</math> must be connected. Therefore, the image of <math>\R</math> under <math>f</math> must be a subset of a component of <math>\R_{\ell}/</math> Since this image is nonempty, the only continuous maps from '<math>\R</math> to <math>\R_{\ell},</math> are the constant maps. In fact, any continuous map from a connected space to a totally disconnected space must be constant.

==Quasicomponents==

Let ''X'' be a topological space.  We define a third relation on ''X'': <math>x \equiv_{qc} y</math> if there is no separation of ''X'' into open sets ''A'' and ''B'' such that ''x'' is an element of ''A'' and ''y'' is an element of ''B''.   This is an equivalence relation on ''X'' and the equivalence class <math>QC_x</math> containing ''x'' is called the '''quasicomponent''' of ''x''.{{sfn|Willard|2004|loc=Problem 26B|pp=195–196}}

<math>QC_x</math> can also be characterized as the intersection of all [[clopen]] subsets of ''X'' that contain ''x''.{{sfn|Willard|2004|loc=Problem 26B|pp=195–196}} Accordingly <math>QC_x</math> is closed; in general it need not be open.

Evidently <math>C_x \subseteq QC_x</math> for all <math>x \in X.</math>{{sfn|Willard|2004|loc=Problem 26B|pp=195–196}} Overall we have the following containments among path components, components and quasicomponents at ''x'':
<math display=block>PC_x \subseteq C_x \subseteq QC_x.</math>

If ''X'' is locally connected, then, as above, <math>C_x</math> is a clopen set containing ''x'', so <math>QC_x \subseteq C_x</math> and thus <math>QC_x = C_x.</math>  Since local path connectedness implies local connectedness, it follows that at all points ''x'' of a locally path connected space we have
<math display=block>PC_x = C_x = QC_x.</math>

Another class of spaces for which the quasicomponents agree with the components is the class of compact Hausdorff spaces.{{sfn|Engelking|1989|loc=Theorem 6.1.23|p=357}}

===Examples===

# An example of a space whose quasicomponents are not equal to its components is a sequence with a double limit point. This space is totally disconnected, but both limit points lie in the same quasicomponent, because any clopen set containing one of them must contain a tail of the sequence, and thus the other point too.
# The space <math>(\{0\}\cup\{\frac{1}{n} : n \in \Z^+\}) \times [-1,1] \setminus \{(0,0)\}</math> is locally compact and Hausdorff but the sets <math>\{0\} \times [-1,0)</math> and <math>\{0\} \times (0,1]</math> are two different components which lie in the same quasicomponent.
# The [[Arens–Fort space]] is not locally connected, but nevertheless the components and the quasicomponents coincide: indeed <math>QC_x = C_x = \{x\}</math> for all points ''x''.{{sfn|Steen| Seebach|1995|pp=54-55}}

==See also==
* {{annotated link|Locally simply connected space}}
* {{annotated link|Semi-locally simply connected}}
* [[MLC conjecture|It is conjectured that the Mandelbrot set is locally connected]]

==Notes==
{{reflist}}

==References==
* {{citation|last=Engelking|first=Ryszard| author-link=Ryszard Engelking|title=General Topology|publisher=Heldermann Verlag, Berlin|year=1989| isbn=3-88538-006-4}}
* {{citation
       | last=Kelley
       | first=John L.
       | author-link=John L. Kelley
       | title=General Topology
       | edition=2nd
       | publisher=Springer-Verlag
       | location=New York
       | year=1975
       | orig-date=1955
       | volume=27
       | series=[[Graduate Texts in Mathematics]]
       | isbn=978-0-387-90125-1
       | oclc=1365153
       | chapter=
       | pp=
       | p=
       }}
* {{citation
       | last=Munkres
       | first=James R.
       | author-link=James Munkres
       | title=Topology
       | edition=2nd
       | publisher=[[Prentice Hall, Inc]]
       | location=[[Upper Saddle River, NJ]]
       | year=2000
       | isbn=978-0-13-181629-9
       | oclc=42683260
       | chapter=
       | pp=
       | p=
       }}
* {{Citation|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]]|orig-year=1978|publisher=Dover Publications, Inc.|location=Mineola, NY|edition=[[Dover Publications|Dover]] reprint of 1978|isbn=978-0-486-68735-3|mr=1382863 |year=1995}}
* {{Willard 2004}}

==Further reading==
* {{Citation|doi=10.1090/S0002-9939-1972-0296913-7|title=Continuous Functions from a Connected Locally Connected Space into a Connected Space with a Dispersion Point|first=C. A.|last= Coppin|journal=Proceedings of the American Mathematical Society|volume=32|issue= 2|year=1972|pages=625–626|jstor=2037874|publisher=American Mathematical Society|doi-access=free}}. For Hausdorff spaces, it is shown that any continuous function from a connected locally connected space into a connected space with a dispersion point is constant
* {{Citation|title=A Note on Connectedness Im Kleinen|first=H. S.|last= Davis|journal=Proceedings of the American Mathematical Society|volume= 19|issue=5|year=1968|pages= 1237–1241|jstor=2036067|publisher=American Mathematical Society|doi=10.1090/s0002-9939-1968-0254814-3|doi-access=free}}.

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