Editing Glossary of general topology (section)

== 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]]