Glossary of Topology

Meagre: If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets. If A is not meagre in X, A is of second category in X.

Metric: See Metric space.

Metric invariant: A metric invariant is a property which is preserved under isometric isomorphism.

Metric map: If X and Y are metric spaces with metrics d_X and d_Y respectively, then a metric map is a function f from X to Y, such that for any points x and y in X, d_Y(f(x), f(y)) ≤ d_X(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.