Editing Glossary of general topology (section)

== P ==<!-- This section is linked from [[Closure (topology)]] -->

;[[Paracompact space|Paracompact]]: A space is [[paracompact space|paracompact]] if every open cover has a locally finite open refinement.  Paracompact implies metacompact.<ref name=ss23>Steen &amp; Seebach (1978) p.23</ref>  Paracompact Hausdorff spaces are normal.<ref name=ss25>Steen &amp; Seebach (1978) p.25</ref>

;[[Partition of unity]]: A partition of unity of a space ''X'' is a set of continuous functions from ''X'' to [0, 1] such that any point has a neighbourhood where all but a [[finite set|finite]] number of the functions are identically zero, and the sum of all the functions on the entire space is identically 1.

;[[Path (topology)|Path]]: A [[Path (topology)|path]] in a space ''X'' is a continuous map ''f'' from the closed unit [[interval (mathematics)|interval]] [0, 1] into ''X''. The point ''f''(0) is the initial point of ''f''; the point ''f''(1) is the terminal point of ''f''.<ref name=ss29>Steen &amp; Seebach (1978) p.29</ref>

;[[Path-connected space|Path-connected]]: A space ''X'' is [[path-connected space|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.<ref name=ss29/>

;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 a set|partition]] of that space, which is [[partition of a set|finer]] than the partition into connected components.<ref name=ss29/> The set of path-connected components of a space ''X'' is denoted [[homotopy groups|π<sub>0</sub>(''X'')]].

;Perfectly normal: a normal space which is also a G<sub>δ</sub>.<ref name=ss162/>

;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''.<ref>Hart, Nagata, Vaughan Sect. d-22, page 227</ref>

;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 (topology)|Closure]]'''.

;[[Polish space|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 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.

;[[Polytopological space]]: A polytopological space is a [[set (mathematics)|set]] <math>X</math> together with a [[family of sets|family]] <math>\{\tau_i\}_{i\in I}</math> of [[Topological_space#Definition_via_open_sets|topologies]] on <math>X</math> that is [[Total order|linearly ordered]] by the [[Subset|inclusion relation]] where <math>I</math> is an arbitrary [[index set]].

;P-point: A point of a topological space is a P-point if its [[Neighbourhood system|filter of neighbourhoods]] is closed under countable intersections.

;Pre-compact: See '''[[Relatively compact]]'''.

;{{visible anchor|Pre-open set|Preopen set}}: A subset ''A'' of a topological space ''X'' is preopen if <math>A \subseteq \operatorname{Int}_X \left( \operatorname{Cl}_X A \right)</math>.{{sfn|Hart|Nagata|Vaughan|2004|pp=8–9}}

;Prodiscrete topology: The prodiscrete topology on a product ''A''<sup>''G''</sup> is the product topology when each factor ''A'' is given the discrete topology.<ref>{{cite book | last1=Ceccherini-Silberstein | first1=Tullio | last2=Coornaert | first2=Michel | title=Cellular automata and groups | zbl=1218.37004 | series=Springer Monographs in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | isbn=978-3-642-14033-4 | year=2010 | page=3 }}</ref>

;[[Product topology]]: If <math>\left(X_i\right)</math> is a collection of spaces and ''X'' is the (set-theoretic) [[Cartesian product]] of <math>\left(X_i\right),</math> 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 <math>f^{-1}(C)</math> is a compact set in ''X'' for any compact subspace ''C'' of ''Y''.

;[[Proximity space]]: A proximity space (''X'',&nbsp;'''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'' is non-empty
:#If ''A'' and ''B'' have non-empty intersection, then ''A'' '''d''' ''B''
:#''A'' '''d''' (''B''&nbsp;<math>\cup</math>&nbsp;''C'') [[if and only if]] (''A'' '''d''' ''B'' or ''A'' '''d''' ''C'')
:#If, for all subsets ''E'' of ''X'', we have (''A'' '''d''' ''E'' or ''B'' '''d''' ''E''), then we must have ''A'' '''d''' (''X'' − ''B'')

;[[Pseudocompact]]: A space is pseudocompact if every [[real number|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 [[Real number|real]]-valued function <math>d : M \times M \to \R</math> 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'', [[Set subtraction|minus]] {''x''}. For instance, the [[interval (mathematics)|interval]] (−1, 1) = {''y'' : −1 < ''y'' < 1} is a neighbourhood of ''x'' = 0 in the [[real line]], so the set <math>(-1, 0) \cup (0, 1) = (-1, 1) - \{ 0 \}</math> is a punctured neighbourhood of 0.