C
- Category of topological spaces
- The category Top has topological spaces as objects and continuous maps as morphisms.
- Cauchy sequence
- A sequence {xn} 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(xm, xn) < 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) := {y in M : d(x, y) ≤ 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 T1 and T2 are topologies on X, then T1 is coarser (or smaller, weaker) than T2 if T1 is contained in T2. 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 neighbourhoods.
- Completely normal Hausdorff
- A completely normal Hausdorff space (or T5 space) is a completely normal T1 space. (A completely normal space is Hausdorff if and only if it is T1, so the terminology is consistent.) Every completely normal Hausdorff space is normal Hausdorff.
- Completely regular
- A space is 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
- See Tychonoff.
- Component
- See Connected component/Path-connected component.
- Connected
- A space is connected if it is not the union of a pair of disjoint nonempty open sets. Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.
- Connected component
- A connected component of a space is a maximal nonempty connected subspace. Each connected component is closed, and the set of connected components of a space is a partition of that space.
- Continuous
- A function from one space to another is continuous if the preimage of every open set is open.
- Continuum
- A space is called a continuum if it a compact, connected Hausdorff space.
- Contractible
- A space X is contractible if the identity map on X is homotopic to a constant map. Every contractible space is simply connected.
- Coproduct topology
- If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous.
- 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 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
- 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.
Read more about this topic: Glossary Of Topology