is non-empty If A and B have non-empty intersection, then A d B A d (B C) if and only if (A d B or A d C) If, for all subsets E of X, we have (A d E or B d E), then we must have A d (X − B) Pseudocompact A space is pseudocompact if every real-valued continuous function on the space is bounded. Pseudometric See Pseudometric space. Pseudometric space A pseudometric space (M, d) is a set M equipped with a real-valued function satisfying all the conditions of a metric space, except possibly the identity of indiscernibles. That is, points in a pseudometric space may be "infinitely close" without being identical. The function d is a pseudometric on M. Every metric is a pseudometric. Punctured neighbourhood/Punctured neighborhood A punctured neighbourhood of a point x is a neighbourhood of x, minus {x}. For instance, the interval (−1, 1) = {y : −1 < y < 1} is a neighbourhood of x = 0 in the real line, so the set is a punctured neighbourhood of 0. Q Quasicompact See compact. Some authors define "compact" to include the Hausdorff separation axiom, and they use the term quasicompact to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French. Quotient map If X and Y are spaces, and if f is a surjection from X to Y, then f is a quotient map (or identification map) if, for every subset U of Y, U is open in Y if and only if f 1(U) is open in X. In other words, Y has the f-strong topology. Equivalently, is a quotient map if and only if it is the transfinite composition of maps , where is a subset. Note that this does not imply that f is an open function. Quotient space If X is a space, Y is a set, and f : X → Y is any surjective function, then the Quotient topology on Y induced by f is the finest topology for which f is continuous. The space X is a quotient space or identification space. By definition, f is a quotient map. The most common example of this is to consider an equivalence relation on X, with Y the set of equivalence classes and f