the natural projection map. This construction is dual to the construction of the subspace topology. R Refinement A cover K is a refinement of a cover L if every member of K is a subset of some member of L. Regular A space is regular if, whenever C is a closed set and x is a point not in C, then C and x have disjoint neighbourhoods. Regular Hausdorff A space is regular Hausdorff (or T3) if it is a regular T0 space. (A regular space is Hausdorff if and only if it is T0, so the terminology is consistent.) A subset of a space X is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior. An example of a non-regular open set is the set U = ∪ in R with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a complete Boolean algebra. Relatively compact A subset Y of a space X is relatively compact in X if the closure of Y in X is compact. Residual If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X. Also called comeagre or comeager. Resolvable A topological space is called resolvable if it is expressible as the union of two disjoint dense subsets. Rim-compact A space is rim-compact if it has a base of open sets whose boundaries are compact. S S-space An S-space is a hereditarily separable space which is not hereditarily Lindelöf. Scattered A space X is scattered if every nonempty subset A of X contains a point isolated in A. Scott The Scott topology on a poset is that in which the open sets are those Upper sets inaccessible by directed joins. Second category See Meagre. Second-countable A space is second-countable or perfectly separable if it has a countable base for its topology. Every second-countable space is first-countable, separable, and Lindelöf. Semilocally simply connected A space X is semilocally simply connected if, for every point x in X, there is a neighbourhood U of x such that every loop at x in U is homotopic in X to the constant loop x. Every simply connected space and every locally