Glossary of Topology - P

P

Paracompact

A space is paracompact if every open cover has a locally finite open refinement. Paracompact implies metacompact. Paracompact Hausdorff spaces are normal.

Partition of unity

A partition of unity of a space X is a set of continuous functions from X to such that any point has a neighbourhood where all but a finite number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.

Path

A path in a space X is a continuous map f from the closed unit interval into X. The point f(0) is the initial point of f; the 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 G_δ.

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 topologically complete, i.e. if it is homeomorphic to a separable and complete metric 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.

Product topology

If {X_i} is a collection of spaces and X is the (set-theoretic) product of {X_i}, 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 f−1(C) is a compact set in X for any compact subspace C of Y.

Proximity space

A proximity space (X, δ) is a set X equipped with a binary relation δ between subsets of X satisfying the following properties:

For all subsets A, B and C of X,

1. A δ B implies B δ A
2. A δ B implies A is non-empty
3. If A and B have non-empty intersection, then A δ B
4. A δ (B ∪ C) iff (A δ B or A δ C)
5. If, for all subsets E of X, we have (A δ E or B δ E), then we must have A δ (X − B)

Pseudocompact

A space is pseudocompact if every real-valued continuous function on the space is bounded.

Pseudometric

See Pseudometric space.

Pseudometric space

A pseudometric space (M, d) is a set M equipped with a function d : M × M → R satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. The function d is a pseudometric on M. Every metric is a pseudometric.

Punctured neighbourhood/Punctured neighborhood

A punctured neighbourhood of a point x is a neighbourhood of x, minus {x}. For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of x = 0 in the real line, so the set (−1, 0) ∪ (0, 1) = (−1, 1) − {0} is a punctured neighbourhood of 0.