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. Cosmic space A continuous image of some separable metric space. 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