metric invariant is a property which is preserved under isometric isomorphism. Metric map If X and Y are metric spaces with metrics dX and dY respectively, then a metric map is a function f from X to Y, such that for any points x and y in X, dY(f(x), f(y)) ≤ dX(x, y). A metric map is strictly metric if the above inequality is strict for all x and y in X. Metric space A metric space (M, d) is a set M equipped with a function d : M × M → R satisfying the following axioms for all x, y, and z in M: d(x, y) ≥ 0 d(x, x) = 0 if d(x, y) = 0 then x = y (identity of indiscernibles) d(x, y) = d(y, x) (symmetry) d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality) The function d is a metric on M, and d(x, y) is the distance between x and y. The collection of all open balls of M is a base for a topology on M; this is the topology on M induced by d. Every metric space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metric space is first-countable. Metrizable/Metrisable A space is metrizable if it is homeomorphic to a metric space. Every metrizable space is Hausdorff and paracompact (and hence normal and Tychonoff). Every metrizable space is first-countable. Monolith Every non-empty ultra-connected compact space X has a largest proper open subset; this subset is called a monolith. Moore space A Moore space is a developable regular Hausdorff space. N Nearly open see preopen. Neighbourhood/Neighborhood A neighbourhood of a point x is a set containing an open set which in turn contains the point x. More generally, a neighbourhood of a set S is a set containing an open set which in turn contains the set S. A neighbourhood of a point x is thus a neighbourhood of the singleton set {x}. (Note that under this definition, the neighbourhood itself need not be open. Many authors require that neighbourhoods be open; be careful to note conventions.) Neighbourhood base/basis See Local base. Neighbourhood system for a point x A neighbourhood system at a point x in a space is the collection of all neighbourhoods of x. Net A net in a space X is a map from a directed set A to X. A net from A to