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- Scott
- The Scott topology on a poset is that in which the open sets are those Upper sets inaccessible by directed joins.
- Second category
- See Meagre.
- Second-countable
- A space is second-countable or perfectly separable if it has a countable base for its topology. 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.)
- Semiregular
- A space is semiregular if the regular open sets form a base.
- Separable
- A space is separable if it has a countable dense subset.
- Separated
- Two sets A and B are separated if each is 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
- A space is simply connected if it is path-connected and every loop is homotopic to a constant map.
- Smaller topology
- See Coarser topology.
- Sober
- In a sober space, every irreducible closed subset is the closure of exactly one point: that is, has a unique generic point.
- Star
- The star of a point in a given cover of a topological space is the union of all the sets in the cover that contain the point. See star refinement.
- -Strong topology
- Let be a map of topological spaces. We say that has the -strong topology if, for every subset, one has that is open in if and only if is open in
- Stronger topology
- See Finer topology. Beware, some authors, especially analysts, 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 a union of finite intersections of sets in the subbase. If B is any collection of subsets of a set X, the topology on X generated by B is the smallest topology containing B; this topology consists of the empty set, X and all unions of finite intersections of elements of B.
- 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
- 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.
Read more about this topic: Glossary Of Topology