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 . Semi-preopen A subset A of a topological space X is called semi-preopen if 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