point f(1) is the terminal point of f. Path-connected A space X is path-connected if, for every two points x, y in X, there is a path f from x to y, i.e., a path with initial point f(0) = x and terminal point f(1) = y. Every path-connected space is connected. Path-connected component A path-connected component of a space is a maximal nonempty path-connected subspace. The set of path-connected components of a space is a partition of that space, which is finer than the partition into connected components. The set of path-connected components of a space X is denoted π0(X). Perfectly normal a normal space which is also a Gδ. π-base A collection B of nonempty open sets is a π-base for a topology τ if every nonempty open set in τ includes a set from B. Point A point is an element of a topological space. More generally, a point is an element of any set with an underlying topological structure; e.g. an element of a metric space or a topological group is also a "point". Point of closure See Closure. Polish A space is Polish if it is separable and completely metrizable, i.e. if it is homeomorphic to a separable and complete metric space. Polyadic A space is polyadic if it is the continuous image of the power of a one-point compactification of a locally compact, non-compact Hausdorff space. P-point A point of a topological space is a P-point if its filter of neighbourhoods is closed under countable intersections. Pre-compact See Relatively compact. A subset A of a topological space X is preopen if . Prodiscrete topology The prodiscrete topology on a product AG is the product topology when each factor A is given the discrete topology. Product topology If is a collection of spaces and X is the (set-theoretic) Cartesian product of then the product topology on X is the coarsest topology for which all the projection maps are continuous. Proper function/mapping A continuous function f from a space X to a space Y is proper if is a compact set in X for any compact subspace C of Y. Proximity space A proximity space (X, d) is a set X equipped with a binary relation d between subsets of X satisfying the following properties: For all subsets A, B and C of X, A d B implies B d A A d B implies A