X carries the discrete topology. Discrete topology See discrete space. Disjoint union topology See Coproduct topology. Dispersion point If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one-point sets). Distance See metric space. Dunce hat (topology) E Entourage See Uniform space. Exterior The exterior of a set is the interior of its complement. F Fσ set An Fσ set is a countable union of closed sets. Filter See also: Filters in topology. A filter on a space X is a nonempty family F of subsets of X such that the following conditions hold: The empty set is not in F. The intersection of any finite number of elements of F is again in F. If A is in F and if B contains A, then B is in F. Final topology On a set X with respect to a family of functions into , is the finest topology on X which makes those functions continuous. Fine topology (potential theory) On Euclidean space , the coarsest topology making all subharmonic functions (equivalently all superharmonic functions) continuous. Finer topology If X is a set, and if T1 and T2 are topologies on X, then T2 is finer (or larger, stronger) than T1 if T2 contains T1. Beware, some authors, especially analysts, use the term weaker. Finitely generated See Alexandrov topology. First category See Meagre. First-countable A space is first-countable if every point has a countable local base. Fréchet See T1. Frontier See Boundary. Full set A compact subset K of the complex plane is called full if its complement is connected. For example, the closed unit disk is full, while the unit circle is not. Functionally separated Two sets A and B in a space X are functionally separated if there is a continuous map f: X → [0, 1] such that f(A) = 0 and f(B) = 1. G Gδ set A Gδ set or inner limiting set is a countable intersection of open sets. Gδ space A space in which every closed set is a Gδ set. Generic point A generic point for a closed set is a point for which the closed set is the closure of the singleton set containing that point. H Hausdorff A Hausdorff space (or T2 space) is