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{{Short description|Topological space characterized by sequences}}
In [[topology]] and related fields of [[mathematics]], a '''sequential space''' is a [[topological space]] whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak [[axiom of countability]], and all [[first-countable space]]s (notably [[metric space]]s) are sequential.

In any topological space <math>(X, \tau),</math> if a convergent sequence is contained in a [[closed set]] <math>C,</math> then the [[Limit (mathematics)|limit]] of that sequence must be contained in <math>C</math> as well. Sets with this property are known as '''sequentially closed'''. Sequential spaces are precisely those topological spaces for which sequentially closed sets are in fact closed. (These definitions can also be rephrased in terms of sequentially open sets; see below.) Said differently, any topology can be described in terms of [[Net (mathematics)|nets]] (also known as Moore–Smith sequences), but those sequences may be "too long" (indexed by too large an ordinal) to compress into a sequence. Sequential spaces are those topological spaces for which nets of countable length (i.e., sequences) suffice to describe the topology.

Any topology can be [[Finer topology|refined]] (that is, made finer) to a sequential topology, called the '''sequential coreflection''' of <math>X.</math>

The related concepts of [[Fréchet–Urysohn space]]s, {{mvar|T}}-sequential spaces, and <math>N</math>-sequential spaces are also defined in terms of how a space's topology interacts with sequences, but have subtly different properties.

Sequential spaces and <math>N</math>-sequential spaces were introduced by [[Stan Franklin|S. P. Franklin]].<ref name="Snipes T-sequential spaces" />

==History==

Although spaces satisfying such properties had implicitly been studied for several years, the first formal definition is due to S. P. Franklin in 1965. Franklin wanted to determine "the classes of topological spaces that can be specified completely by the knowledge of their convergent sequences", and began by investigating the [[first-countable space]]s, for which it was already known that sequences sufficed. Franklin then arrived at the modern definition by abstracting the necessary properties of first-countable spaces.

==Preliminary definitions==

{{See also|Filters in topology|Net (mathematics)}}

Let <math>X</math> be a set and let <math>x_{\bull} = \left(x_i\right)_{i=1}^{\infty}</math> be a [[sequence]] in <math>X</math>; that is, a family of elements of <math>X</math>, [[indexed family|indexed]] by the [[natural numbers]]. In this article, <math>x_{\bull} \subseteq S</math> means that each element in the sequence <math>x_{\bull}</math> is an element of <math>S,</math> and, if <math>f : X \to Y</math> is a map, then <math>f\left(x_{\bull}\right) = \left(f\left(x_i\right)\right)_{i=1}^{\infty}.</math> For any index <math>i,</math> the tail of <math>x_{\bull}</math> starting at <math>i</math> is the sequence <math display="block">x_{\geq i} = (x_i, x_{i+1}, x_{i+2}, \ldots)\text{.}</math> A sequence <math>x_{\bull}</math> is eventually in <math>S</math> if some tail of <math>x_{\bull}</math> satisfies <math>x_{\geq i} \subseteq S.</math>

Let <math>\tau</math> be a [[topological space|topology]] on <math>X</math> and <math>x_{\bull}</math> a sequence therein. The sequence <math>x_{\bull}</math> [[Convergent sequence|converges]] to a point <math>x \in X,</math> written <math>x_{\bull}\overset{\tau}{\to} x</math> (when context allows, <math>x_\bull\to x</math>), if, for every neighborhood <math>U\in\tau</math> of <math>x,</math> eventually <math>x_{\bull}</math> is in <math>U.</math> <math>x</math> is then called a limit point of <math>x_{\bull}.</math>

A function <math>f : X \to Y</math> between topological spaces is [[sequentially continuous]] if <math>x_\bull\to x</math> implies <math>f(x_\bull)\to f(x).</math>

== Sequential closure/interior ==
Let <math>(X, \tau)</math> be a topological space and let <math>S \subseteq X</math> be a subset. The [[Closure (topology)|topological closure]] (resp. [[Interior (topology)|topological interior]]) of <math>S</math> in <math>(X, \tau)</math> is denoted by <math>\operatorname{cl}_X S</math> (resp. <math>\operatorname{int}_X S</math>).

The '''sequential closure''' of <math>S</math> in <math>(X, \tau)</math> is the set<math display="block">\operatorname{scl}(S) = \left\{x \in X: \text{there exists a sequence }s_{\bull} \subseteq S\text{ such that }s_{\bull} \to x \right\}</math>which defines a map, the '''sequential closure operator''', on the power set of <math>X.</math> If necessary for clarity, this set may also be written <math>\operatorname{scl}_{X}(S)</math> or <math>\operatorname{scl}_{(X,\tau)}(S).</math> It is always the case that <math>\operatorname{scl}_X S \subseteq \operatorname{cl}_X S,</math> but the reverse may fail.

The '''sequential interior''' of <math>S</math> in <math>(X, \tau)</math> is the set<math display="block">\operatorname{sint}(S) = \{s \in S: \text{whenever }x_{\bull}\subseteq X\text{ and }x_{\bull}\to s,\text{ then }x_{\bull}\text{ is eventually in }S\}</math>(the topological space again indicated with a subscript if necessary).

Sequential closure and interior satisfy many of the nice properties of ''topological'' closure and interior: for all subsets <math>R, S \subseteq X,</math>

<ul>
<li><math>\operatorname{scl}_X(X\setminus S)=X\setminus\operatorname{sint}_X(S)</math> and <math>\operatorname{sint}_X(X\setminus S)=X\setminus\operatorname{scl}_X(S)</math>;
{| class="toccolours collapsible collapsed" width="60%" style="text-align:left"
! Proof
|-
| <p>Fix <math>x\in\operatorname{sint}(X\setminus S).</math> If <math>x\in\operatorname{scl}(S),</math> then there exists <math>s_\bull\subseteq S</math> with <math>s_\bull\to x.</math> But by the definition of sequential interior, eventually <math>s_\bull</math> is in <math>X\setminus S,</math> contradicting <math>s_\bull\subseteq S.</math> </p>

Conversely, suppose <math>x\notin\operatorname{sint}(X\setminus S)</math>; then there exists a sequence <math>s_\bull\subseteq X</math> with <math>s_\bull\to x</math> that is not eventually in <math>X\setminus S.</math> By passing to the subsequence of elements not in <math>X\setminus S,</math> we may assume that <math>s_\bull\subseteq S.</math> But then <math>x\in\operatorname{scl}(S).</math> {{align|right|▮}}

|}</li>
<li><math>\operatorname{scl}(\emptyset) = \emptyset</math> and <math>\operatorname{sint}(\emptyset)=\emptyset</math>;</li>
<li><math display="inline">\operatorname{sint}(S)\subseteq S\subseteq\operatorname{scl}(S)</math>;</li>
<li><math>\operatorname{scl}(R\cup S)=\operatorname{scl}(R)\cup\operatorname{scl}(S)</math>; and</li>
<li><math display="inline">\operatorname{scl}(S)\subseteq\operatorname{scl}(\operatorname{scl}(S)).</math></li>
</ul>

That is, sequential closure is a [[preclosure operator]]. Unlike topological closure, sequential closure is not [[Idempotence|idempotent]]: the last containment may be strict. Thus sequential closure is not a ([[Kuratowski closure operator|Kuratowski]]) [[closure operator]].

===Sequentially closed and open sets===
{{anchor|Sequentially open|Sequentially closed}}

A set <math>S</math> is sequentially closed if <math>S=\operatorname{scl}(S)</math>; equivalently, for all <math>s_{\bull}\subseteq S</math> and <math>x \in X</math> such that <math>s_{\bull}\overset{\tau}{\to}x,</math> we must have <math>x\in S.</math><ref group="note">You cannot simultaneously apply this "test" to infinitely many subsets (for example, you can not use something akin to the [[axiom of choice]]). Not all sequential spaces are [[Fréchet-Urysohn space|Fréchet-Urysohn]], but only in those spaces can the closure of a set <math>S</math> can be determined without it ever being necessary to consider any set other than <math>S.</math></ref>

A set <math>S</math> is defined to be sequentially open if its [[Complement (set theory)|complement]] is sequentially closed. Equivalent conditions include:

<ul>
<li><math>S = \operatorname{sint}(S)</math> or</li>
<li>For all <math>x_{\bull}\subseteq X</math> and <math>s \in S</math> such that <math>x_{\bull}\overset{\tau}{\to}s,</math> eventually <math>x_{\bull}</math> is in <math>S</math> (that is, there exists some integer <math>i</math> such that the tail <math>x_{\geq i} \subseteq S</math>).</li>
</ul>

A set <math>S</math> is a '''sequential neighborhood''' of a point <math>x \in X</math> if it contains <math>x</math> in its sequential interior; sequential neighborhoods need ''not'' be sequentially open (see {{Slink||T- and N-sequential spaces}} below).

It is possible for a subset of <math>X</math> to be sequentially open but not open. Similarly, it is possible for there to exist a sequentially closed subset that is not closed.

==Sequential spaces and coreflection==
As discussed above, sequential closure is not in general idempotent, and so not the closure operator of a topology. One can obtain an idempotent sequential closure via [[transfinite iteration]]: for a [[successor ordinal]] <math>\alpha+1,</math> define (as usual)<math display="block">(\operatorname{scl})^{\alpha+1}(S)=\operatorname{scl}((\operatorname{scl})^\alpha(S))</math>and, for a [[limit ordinal]] <math>\alpha,</math> define<math display="block">(\operatorname{scl})^\alpha(S)=\bigcup_{\beta<\alpha}{(\operatorname{scl})^\beta(S)}\text{.}</math>This process gives an ordinal-indexed increasing sequence of sets; as it turns out, that sequence always stabilizes by index <math>\omega_1</math> (the [[first uncountable ordinal]]). Conversely, the '''sequential order''' of <math>X</math> is the minimal ordinal at which, for any choice of <math>S,</math> the above sequence will stabilize.<ref>*{{cite journal |last1=Arhangel'skiĭ |first1=A. V. |last2=Franklin |first2=S. P. |year=1968 |title=Ordinal invariants for topological spaces. |journal=Michigan Math. J. |volume=15 |issue=3 |pages=313–320 |doi=10.1307/mmj/1029000034 |doi-access=free}}</ref>

The '''transfinite sequential closure''' of <math>S</math> is the terminal set in the above sequence: <math>(\operatorname{scl})^{\omega_1}(S).</math> The operator <math>(\operatorname{scl})^{\omega_1}</math> is idempotent and thus a [[closure operator]]. In particular, it defines a topology, the sequential coreflection. In the sequential coreflection, every sequentially-closed set is closed (and every sequentially-open set is open).<ref>{{Cite journal |last=Baron |first=S. |date=October 1968 |title=The Coreflective Subcategory of Sequential Spaces |journal=[[Canadian Mathematical Bulletin]] |language=en |volume=11 |issue=4 |pages=603–604 |doi=10.4153/CMB-1968-074-4 |s2cid=124685527 |issn=0008-4395|doi-access=free }}</ref>

=== Sequential spaces ===
A topological space <math>(X, \tau)</math> is '''sequential''' if it satisfies any of the following equivalent conditions:
<ul>
<li><math>\tau</math> is its own sequential coreflection.<ref>{{cite web |title=Topology of sequentially open sets is sequential? |url=https://math.stackexchange.com/questions/3737020 |website=Mathematics Stack Exchange}}</ref></li>
<li>Every sequentially open subset of <math>X</math> is open.</li>
<li>Every sequentially closed subset of <math>X</math> is closed.</li>
<li>For any subset <math>S \subseteq X</math> that is {{em|not}} closed in <math>X,</math> there exists some<ref group="note">A [[Fréchet–Urysohn space]] is defined by the analogous condition for ''all'' (not "some") such <math>x</math>: <blockquote>For any subset <math>S \subseteq X</math> that is not closed in <math>X,</math> ''for any'' <math>x \in \operatorname{cl}_X(S) \setminus S,</math> there exists a sequence in <math>S</math> that converges to <math>x.</math></blockquote></ref> <math>x\in\operatorname{cl}(S)\setminus S</math> and a sequence in <math>S</math> that converges to <math>x.</math><ref name="Arkhangel'skii, A.V. and Pontryagin L.S.">Arkhangel'skii, A.V. and Pontryagin L.S.,{{pad|1px}} General Topology I, definition 9 p.12</ref>
</li>
<li>(Universal Property) For every topological space <math>Y,</math> a map <math>f : X \to Y</math> is [[Continuous function (topology)|continuous]] if and only if it is [[Sequential continuity|sequentially continuous]] (if <math>x_{\bull} \to x</math> then <math>f\left(x_{\bull}\right) \to f(x)</math>).<ref>{{Cite journal |last1=Baron |first1=S. |last2=Leader |first2=Solomon |date=1966 |title=Solution to Problem #5299 |url=https://www.jstor.org/stable/2314834 |journal=The American Mathematical Monthly |volume=73 |issue=6 |pages=677–678 |doi=10.2307/2314834 |jstor=2314834 |issn=0002-9890}}</ref> </li>
<li><math>X</math> is the quotient of a first-countable space.</li>
<li><math>X</math> is the quotient of a metric space.</li>
</ul>

By taking <math>Y = X</math> and <math>f</math> to be the identity map on <math>X</math> in the universal property, it follows that the class of sequential spaces consists precisely of those spaces whose topological structure is determined by convergent sequences. If two topologies agree on convergent sequences, then they necessarily have the same sequential coreflection. Moreover, a function from <math>Y</math> is sequentially continuous if and only if it is continuous on the sequential coreflection (that is, when pre-composed with <math>f</math>).

== {{mvar|T}}- and {{Mvar|N}}-sequential spaces ==
A '''{{mvar|T}}-sequential space''' is a topological space with sequential order 1, which is equivalent to any of the following conditions:<ref name="Snipes T-sequential spaces">{{Cite journal |last=Snipes |first=Ray |date=1972 |title=T-sequential topological spaces |url=http://matwbn.icm.edu.pl/ksiazki/fm/fm77/fm7719.pdf |journal=Fundamenta Mathematicae |language=en |volume=77 |issue=2 |pages=95–98 |doi=10.4064/fm-77-2-95-98 |issn=0016-2736}}</ref> <ul>
<li>The sequential closure (or interior) of every subset of <math>X</math> is sequentially closed (resp. open).</li>
<li><math>\operatorname{scl}</math> or <math>\operatorname{sint}</math> are idempotent.</li>
<li><math display="inline">\operatorname{scl}(S)=\bigcap_{\text{sequentially closed }C\supseteq S}{C}</math> or <math display="inline">\operatorname{sint}(S)=\bigcup_{\text{sequentially open }U\subseteq S}{U}</math>
</li>
<li>Any sequential neighborhood of <math>x \in X</math> can be shrunk to a sequentially-open set that contains <math>x</math>; formally, sequentially-open neighborhoods are a [[neighborhood basis]] for the sequential neighborhoods.</li>
<li>For any <math>x \in X</math> and any sequential neighborhood <math>N</math> of <math>x,</math> there exists a sequential neighborhood <math>M</math> of <math>x</math> such that, for every <math>m \in M,</math> the set <math>N</math> is a sequential neighborhood of <math>m.</math>
</li>
</ul>

Being a {{mvar|T}}-sequential space is incomparable with being a sequential space; there are sequential spaces that are not {{mvar|T}}-sequential and vice-versa. However, a topological space <math>(X, \tau)</math> is called a '''<math>N</math>-sequential''' (or '''neighborhood-sequential''') if it is both sequential and {{mvar|T}}-sequential. An equivalent condition is that every sequential neighborhood contains an open (classical) neighborhood.<ref name="Snipes T-sequential spaces" />

Every [[first-countable space]] (and thus every [[metrizable space]]) is <math>N</math>-sequential. There exist [[topological vector spaces]] that are sequential but {{em|not}} <math>N</math>-sequential (and thus not {{mvar|T}}-sequential).<ref name="Snipes T-sequential spaces" />

===Fréchet–Urysohn spaces===
{{Main|Fréchet–Urysohn space}}

A topological space <math>(X, \tau)</math> is called [[Fréchet–Urysohn space|Fréchet–Urysohn]] if it satisfies any of the following equivalent conditions: <ul>
<li><math>X</math> is hereditarily sequential; that is, every topological subspace is sequential.
</li>
<li>For every subset <math>S \subseteq X,</math> <math>\operatorname{scl}_X S = \operatorname{cl}_X S.</math>
</li>
<li>For any subset <math>S \subseteq X</math> that is not closed in <math>X</math> and every <math>x \in \left(\operatorname{cl}_X S\right) \setminus S,</math> there exists a sequence in <math>S</math> that converges to <math>x.</math>
</li>
</ul>

Fréchet–Urysohn spaces are also sometimes said to be "Fréchet," but should be confused with neither [[Fréchet space]]s in [[functional analysis]] nor the [[T1 space|T<sub>1</sub> condition]].

==Examples and sufficient conditions==

Every [[CW-complex]] is sequential, as it can be considered as a quotient of a metric space.

The [[prime spectrum]] of a commutative [[Noetherian ring]] with the [[Zariski topology]] is sequential.<ref>{{cite web |date=2004 |title=On sequential properties of Noetherian topological spaces |url=http://topology.nipissingu.ca/tp/reprints/v28/tp28210.pdf |access-date=30 Jul 2023}}</ref>

Take the real line <math>\R</math> and [[Quotient space (topology)|identify]] the set <math>\Z</math> of integers to a point. As a quotient of a metric space, the result is sequential, but it is not first countable.

Every [[first-countable space]] is Fréchet–Urysohn and every Fréchet-Urysohn space is sequential. Thus every metrizable or [[Pseudometrizable space|pseudometrizable]] space&nbsp;&mdash; in particular, every [[second-countable space]], [[metric space]], or [[discrete space]]&nbsp;&mdash; is sequential.

Let <math>\mathcal{F}</math> be a set of maps from [[Fréchet–Urysohn space|Fréchet–Urysohn]] spaces to <math>X.</math> Then the [[final topology]] that <math>\mathcal{F}</math> induces on <math>X</math> is sequential.

A Hausdorff [[topological vector space]] is sequential if and only if there exists no strictly finer topology with the same convergent sequences.{{sfn|Wilansky|2013|p=224}}<ref name="Dudley on conv 1964">Dudley, R. M., On sequential convergence - Transactions of the American Mathematical Society Vol 112, 1964, pp. 483-507</ref>

===Spaces that are sequential but not Fréchet-Urysohn===
[[Schwartz space]] <math>\mathcal{S}\left(\R^n\right)</math>and the space <math>C^{\infty}(U)</math> of [[smooth function]]s, as discussed in the article on [[Distribution (mathematics)|distributions]], are both widely-used sequential spaces.<ref name=":0">{{Cite journal
| arxiv=1702.07867
| last1=Gabrielyan | first1=Saak
| title=Topological properties of strict <math>(LF)</math>-spaces and strong duals of Montel strict <math>(LF)</math>-spaces
| journal=Monatshefte für Mathematik
| volume=189
| issue=1
| pages=91–99
| date=2019
| doi=10.1007/s00605-018-1223-6}}</ref><ref name="Shirai 1959">T. Shirai, Sur les Topologies des Espaces de L. Schwartz, Proc. Japan Acad. 35 (1959), 31-36.</ref>

More generally, every infinite-dimensional [[Montel space|Montel]] [[DF-space]] is sequential but not [[Fréchet–Urysohn space|Fréchet–Urysohn]].

Arens' space is sequential, but not Fréchet–Urysohn.<ref>Engelking 1989, Example 1.6.19</ref><ref>{{cite web |last=Ma |first=Dan |date=19 August 2010 |title=A note about the Arens' space |url=http://dantopology.wordpress.com/2010/08/18/a-note-about-the-arens-space/ |access-date=1 August 2013}}</ref>

===Non-examples (spaces that are not sequential)===
The simplest space that is not sequential is the [[cocountable topology]] on an uncountable set. Every convergent sequence in such a space is eventually constant; hence every set is sequentially open. But the cocountable topology is not [[discrete space|discrete]]. (One could call the topology "sequentially discrete".)<ref>{{Cite web |last1=math |last2=Sleziak |first2=Martin |date=Dec 6, 2016 |title=Example of different topologies with same convergent sequences |url=https://math.stackexchange.com/questions/76691/example-of-different-topologies-with-same-convergent-sequences |access-date=2022-06-27 |website=Mathematics Stack Exchange |publisher=StackOverflow |language=en}}</ref>

Let <math>C_c^k(U)</math> denote the [[Distribution (mathematics)|space of]] <math>k</math>[[Distribution (mathematics)|-smooth test functions]] with its canonical topology and let <math>\mathcal{D}'(U)</math> denote the space of distributions, the [[strong dual space]] of <math>C_c^{\infty}(U)</math>; neither are sequential (nor even an [[Ascoli space]]).<ref name=":0" /><ref name="Shirai 1959" /> On the other hand, both <math>C_c^{\infty}(U)</math> and <math>\mathcal{D}'(U)</math> are [[Montel space|Montel]] spaces<ref name="Encyc. Math TVS">{{cite web |author=<!--Not stated--> |date= |title=Topological vector space |url=https://encyclopediaofmath.org/wiki/Topological_vector_space |access-date=September 6, 2020 |website=Encyclopedia of Mathematics |quote="It is a Montel space, hence paracompact, and so normal."}}</ref> and, in the [[Continuous dual space|dual space]] of any Montel space, a ''sequence'' of continuous linear functionals converges in the [[Strong dual space|strong dual topology]] if and only if it converges in the [[weak* topology]] (that is, converges pointwise).<ref name=":0" />{{sfn|Trèves|2006|pp=351-359}}

==Consequences==
Every sequential space has [[countable tightness]] and is [[Compactly generated space|compactly generated]].

If <math>f : X \to Y</math> is a continuous [[open map|open]] [[Surjective map|surjection]] between two Hausdorff sequential spaces then the set <math>\{y:{|f^{-1}(y)| = 1}\}\subseteq Y</math> of points with unique preimage is closed. (By continuity, so is its preimage in <math>X,</math> the set of all points on which <math>f</math> is injective.)

If <math>f : X \to Y</math> is a surjective map (not necessarily continuous) onto a Hausdorff sequential space <math>Y</math> and <math>\mathcal{B}</math> [[Basis (topology)|bases]] for the topology on <math>X,</math> then <math>f : X \to Y</math> is an [[open map]] if and only if, for every <math>x \in X,</math> basic neighborhood <math>B \in \mathcal{B}</math> of <math>x,</math> and sequence <math>y_{\bull} = \left(y_i\right)_{i=1}^{\infty} \to f(x)</math> in <math>Y,</math> there is a subsequence of <math>y_\bull</math> that is eventually in&nbsp;<math>f(B).</math>

==Categorical properties==

The [[full subcategory]] '''Seq''' of all sequential spaces is closed under the following operations in the [[Category (mathematics)|category]] '''Top''' of topological spaces:
{{collist|
* Quotients
* Continuous closed or open [[Image (mathematics)|images]]
* Sums
* [[Limit (category theory)|Inductive limits]]{{disputed inline|Categorical properties|date=March 2019}}
* Open and closed [[Subspace topology|subspaces]]
}}
The category '''Seq''' is {{em|not}} closed under the following operations in '''Top''':
{{collist|
* Continuous images
* Subspaces
* Finite [[Product (category theory)|products]]
}}
Since they are closed under topological sums and quotients, the sequential spaces form a [[coreflective subcategory]] of the [[category of topological spaces]]. In fact, they are the coreflective hull of [[metrizable space]]s (that is, the smallest class of topological spaces closed under sums and quotients and containing the metrizable spaces).

The subcategory '''Seq''' is a [[Cartesian closed category]] with respect to its own product (not that of '''Top'''). The [[exponential object]]s are equipped with the (convergent sequence)-open topology.

P.I. Booth and A. Tillotson have shown that '''Seq''' is the smallest Cartesian closed subcategory of '''Top''' containing the underlying topological spaces of all [[metric space]]s, [[CW-complex]]es, and [[differentiable manifold]]s and that is closed under colimits, quotients, and other "certain reasonable identities" that [[Norman Steenrod]] described as "convenient".<ref name="Steenrod1967">{{harvnb|Steenrod|1967|p=}}</ref>

Every sequential space is [[Compactly generated space|compactly generated]], and finite products in '''Seq''' coincide with those for compactly generated spaces, since products in the category of compactly generated spaces preserve quotients of metric spaces.

==See also==

* {{annotated link|Axiom of countability}}
* {{annotated link|Closed graph property}}
* {{annotated link|First-countable space}}
* {{annotated link|Fréchet–Urysohn space}}
* {{annotated link|Sequence covering map}}

==Notes==
{{reflist|group=note}}

==Citations==
{{reflist}}

==References==

* Arkhangel'skii, A.V. and Pontryagin, L.S., ''General Topology I'', Springer-Verlag, New York (1990) {{isbn|3-540-18178-4}}.
* {{cite journal|last1=Arkhangel'skii|first1=A V|title=Mappings and spaces|journal=Russian Mathematical Surveys|volume=21|issue=4|year=1966|pages=115–162|issn=0036-0279|doi=10.1070/RM1966v021n04ABEH004169|bibcode=1966RuMaS..21..115A|s2cid=250900871 |url=http://www.mathnet.ru/links/0411dc60fab54ffac1cb8172e57c8f69/rm5901.pdf|access-date=10 February 2021}} <!--<ref name="Arkhangel'skii1966">{{harvnb|Arkhangel'skii|1966|p=}}</ref>-->
* {{cite journal|last1=Akiz|first1=Hürmet Fulya|last2=Koçak|first2=Lokman|title=Sequentially Hausdorff and full sequentially Hausdorff spaces|journal=Communications Faculty of Science University of Ankara Series A1Mathematics and Statistics|volume=68|issue=2|year=2019|pages=1724–1732|issn=1303-5991|doi=10.31801/cfsuasmas.424418|url=https://dergipark.org.tr/en/download/article-file/692156|access-date=10 February 2021|doi-access=free}} <!--<ref name="AkizKoçak2019">{{harvnb|Akiz|Koçak|2019|p=}}</ref>-->
* {{cite journal|last1=Boone|first1=James|title=A note on mesocompact and sequentially mesocompact spaces|journal=Pacific Journal of Mathematics|volume=44|issue=1|year=1973|pages=69–74|issn=0030-8730|doi=10.2140/pjm.1973.44.69|doi-access=free}} <!---->
* {{cite journal|last1=Booth|first1=Peter|last2=Tillotson|first2=J.|title=Monoidal closed, Cartesian closed and convenient categories of topological spaces|journal=Pacific Journal of Mathematics|volume=88|issue=1|year=1980|pages=35–53|issn=0030-8730|doi=10.2140/pjm.1980.88.35|url=http://projecteuclid.org/euclid.pjm/1102779712|access-date=10 February 2021|doi-access=free}} <!--<ref name="BoothTillotson1980">{{harvnb|Booth|Tillotson|1973|p=1980}}</ref>-->
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[[Category:General topology]]
[[Category:Properties of topological spaces]]