by or . 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 {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