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. T T0 A space is T0 (or Kolmogorov) if for every pair of distinct points x and y in the space, either there is an open set containing x but not y, or there is an open set containing y but not x. T1 A space is T1 (or Fréchet or accessible) if for every pair of distinct points x and y in the space, there is an open set containing x but not y. (Compare with T0; here, we are allowed to specify which point will be contained in the open set.) Equivalently, a space is T1 if all its singletons are closed. Every T1 space is T0. T2 See Hausdorff space. T3 See Regular Hausdorff. T3½ See Tychonoff space. T4 See Normal Hausdorff. T5 See Completely normal Hausdorff. Top See Category of topological spaces. θ-cluster point, θ-closed, θ-open A point x of a topological space X is a θ-cluster point of a subset A if for every open neighborhood U of x in X. The subset A is θ-closed if it is equal to the set of its θ-cluster points, and θ-open if its complement is θ-closed. Topological invariant A topological invariant is a property which is preserved under homeomorphism. For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces. Topological space A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms: The empty set and X are in T. The union of any collection of sets in T is also in T. The intersection of any pair of sets in T is also in T. The collection T is a topology on X. Topological sum See Coproduct topology. Topologically complete Completely metrizable spaces (i. e. topological spaces homeomorphic to