Editing Glossary of general topology (section)

== C ==

;[[Category of topological spaces]]: The [[category theory|category]] '''[[Category of topological spaces|Top]]''' has [[topological space]]s as [[object (category theory)|objects]] and [[continuous map]]s as [[morphism]]s.

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

;[[Clopen set]]: A set is [[clopen set|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'') := {''y'' in ''M'' : ''d''(''x'', ''y'') ≤ ''r''}, where ''x'' is in ''M'' and ''r'' is a [[positive number|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 (topology)|closure]] of the open ball ''B''(''x''; ''r'').

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

;[[Closed function]]: A function from one space to another is closed if the [[image (mathematics)|image]] of every closed set is closed.

;[[Closure (topology)|Closure]]: The [[closure (topology)|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''<sub>1</sub> and ''T''<sub>2</sub> are topologies on ''X'', then ''T''<sub>1</sub> is [[coarser topology|coarser]] (or '''smaller''', '''weaker''') than ''T''<sub>2</sub> if ''T''<sub>1</sub> is contained in ''T''<sub>2</sub>. Beware, some authors, especially [[mathematical analysis|analyst]]s, use the term '''stronger'''.

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

;[[Compact space|Compact]]: A space is [[compact space|compact]] if every open cover has a [[finite set|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 space|Complete]]: A metric space is [[complete space|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 sets|disjoint]] neighbourhoods.

;Completely normal Hausdorff: A completely normal Hausdorff space (or [[T5 space|'''T<sub>5</sub>''' space]]) is a completely normal T<sub>1</sub> space. (A completely normal space is Hausdorff [[if and only if]] it is T<sub>1</sub>, so the terminology is [[consistent]].) Every completely normal Hausdorff space is normal Hausdorff.

;[[Completely regular space|Completely regular]]: A space is [[Completely regular space|completely regular]] if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and {''x''} are functionally separated.

;[[Completely T3 space|Completely T<sub>3</sub>]]: See '''[[Tychonoff space|Tychonoff]]'''.

;Component: See '''[[Connected space|Connected component]]'''/'''Path-connected component'''.

;[[Connected (topology)|Connected]]: A space is [[connected (topology)|connected]] if it is not the union of a pair of [[Disjoint sets|disjoint]] nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.

;[[connected space|Connected component]]: A [[connected space|connected component]] of a space is a [[maximal set|maximal]] nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a [[partition of a set|partition]] of that space.

;[[Continuity (topology)|Continuous]]: A function from one space to another is [[continuity (topology)|continuous]] if the [[preimage]] of every open set is open.

;[[Continuum (topology)|Continuum]]: A space is called a continuum if it a compact, connected Hausdorff space.

;[[Contractible space|Contractible]]: A space ''X'' is contractible if the [[identity function|identity map]] on ''X'' is homotopic to a constant map. Every contractible space is simply connected.

;[[Coproduct topology]]: If {''X''<sub>''i''</sub>} is a collection of spaces and ''X'' is the (set-theoretic) [[disjoint union]] of {''X''<sub>''i''</sub>}, then the coproduct topology (or '''disjoint union topology''', '''topological sum''' of the ''X''<sub>''i''</sub>) on ''X'' is the finest topology for which all the injection maps are continuous.

;[[Core-compact space]]

;[[Cosmic space]]: A [[continuous function|continuous]] [[Image (mathematics)|image]] of some [[Separable space|separable]] [[metric space]].<ref>{{cite book | title=Encyclopedia of Distances | first1=Michel Marie | last1=Deza | first2=Elena | last2=Deza |author2-link=Elena Deza| publisher=[[Springer-Verlag]] | year=2012 | isbn=978-3642309588 | page=64 }}</ref>

;[[Countable chain condition]]: A space ''X'' satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.

;[[Countably compact]]: A space is countably compact if every [[countable]] open cover has a [[finite set|finite]] subcover. Every countably compact space is pseudocompact and weakly countably compact.

;Countably locally finite:  A collection of subsets of a space ''X'' is '''countably locally finite''' (or '''σ-locally finite''') if it is the union of a [[countable]] collection of locally finite collections of subsets of ''X''.

;[[Cover (topology)|Cover]]: A collection of subsets of a space is a cover (or '''covering''') of that space if the union of the collection is the whole space.

;Covering: See '''Cover'''.

;Cut point: If ''X'' is a connected space with more than one point, then a point ''x'' of ''X'' is a cut point if the subspace ''X'' − {''x''} is disconnected.