P-point

This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.

All spaces in this glossary are assumed to be topological spaces unless stated otherwise.

A

;Absolutely closed: See ''H-closed''

;Accessible: See $T_1$.

;Accumulation point: See limit point.

;Alexandrov topology: The topology of a space ''X'' is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in ''X'' are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.

;Almost discrete: A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated zero-dimensional spaces.

;α-closed, α-open: A subset ''A'' of a topological space ''X'' is α-open if $A \subseteq \operatorname_X \left( \operatorname_X \left( \operatorname_X A \right) \right)$, and the complement of such a set is α-closed.

;Approach space: An approach space is a generalization of metric space based on point-to-set distances, instead of point-to-point.

B

;Baire space: This has two distinct common meanings:
:#A space is a Baire space if the intersection of any countable collection of dense open sets is dense; see Baire space.
:#Baire space is the set of all functions from the natural numbers to the natural numbers, with the topology of pointwise convergence; see Baire space (set theory).

; Base: A collection ''B'' of open sets is a base (or basis) for a topology $\tau$ if every open set in $\tau$ is a union of sets in $B$. The topology $\tau$ is the smallest topology on $X$ containing $B$ and is said to be generated by $B$.

;Basis: See Base.

;β-open: See ''Semi-preopen''.

;b-open, b-closed: A subset $A$ of a topological space $X$ is b-open if $A \subseteq \operatorname_X \left( \operatorname_X A \right) \cup \operatorname_X \left( \operatorname_X A \right)$. The complement of a b-open set is b-closed.

;Borel algebra: The Borel algebra on a topological space $(X,\tau)$ is the smallest $\sigma$-algebra containing all the open sets. It is obtained by taking intersection of all $\sigma$-algebras on $X$ containing $\tau$.

;Borel set: A Borel set is an element of a Borel algebra.

;Boundary: The boundary (or frontier) of a set is the set's closure minus its interior. Equivalently, the boundary of a set is the intersection of its closure with the closure of its complement. Boundary of a set $A$ is denoted by $\partial A$ or $bd$ $A$.

; Bounded: A set in a metric space is bounded if it has finite diameter. Equivalently, a set is bounded if it is contained in some open ball of finite radius. A function taking values in a metric space is bounded if its image is a bounded set.

C

;Category of topological spaces: The category Top has topological spaces as objects and continuous maps as morphisms.

;Cauchy sequence: A sequence in a metric space (''M'', ''d'') is a Cauchy sequence if, for every positive real number ''r'', there is an integer ''N'' such that for all integers ''m'', ''n'' > ''N'', we have ''d''(''x''_''m'', ''x''_''n'') < ''r''.

;Clopen set: A set is clopen if it is both open and closed.

;Closed ball: If (''M'', ''d'') is a metric space, a closed ball is a set of the form ''D''(''x''; ''r'') := , where ''x'' is in ''M'' and ''r'' is a positive real number, the radius of the ball. A closed ball of radius ''r'' is a closed ''r''-ball. Every closed ball is a closed set in the topology induced on ''M'' by ''d''. Note that the closed ball ''D''(''x''; ''r'') might not be equal to the closure of the open ball ''B''(''x''; ''r'').

;Closed set: A set is closed if its complement is a member of the topology.

;Closed function: A function from one space to another is closed if the image of every closed set is closed.

; Closure: The closure of a set is the smallest closed set containing the original set. It is equal to the intersection of all closed sets which contain it. An element of the closure of a set ''S'' is a point of closure of ''S''.

;Closure operator: See Kuratowski closure axioms.

;Coarser topology: If ''X'' is a set, and if ''T''₁ and ''T''₂ are topologies on ''X'', then ''T''₁ is coarser (or smaller, weaker) than ''T''₂ if ''T''₁ is contained in ''T''₂. Beware, some authors, especially analysts, use the term stronger.

;Comeagre: A subset ''A'' of a space ''X'' is comeagre (comeager) if its complement ''X''\''A'' is meagre. Also called residual.

;Compact: A space is compact if every open cover has a finite subcover. Every compact space is Lindelöf and paracompact. Therefore, every compact Hausdorff space is normal. See also quasicompact.

;Compact-open topology: The compact-open topology on the set ''C''(''X'', ''Y'') of all continuous maps between two spaces ''X'' and ''Y'' is defined as follows: given a compact subset ''K'' of ''X'' and an open subset ''U'' of ''Y'', let ''V''(''K'', ''U'') denote the set of all maps ''f'' in ''C''(''X'', ''Y'') such that ''f''(''K'') is contained in ''U''. Then the collection of all such ''V''(''K'', ''U'') is a subbase for the compact-open topology.

;Complete: A metric space is complete if every Cauchy sequence converges.

;Completely metrizable/completely metrisable: See complete space.

;Completely normal: A space is completely normal if any two separated sets have disjoint

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