Metric and Normed Spaces

In this section we recall some basic topological concepts and introduce a number of examples of metric spaces.

A metric space is a set equipped with a metric. metric or distance function is a map with the properties

A semimetric satisfies (i) and (ii), but not necessarily (iii). An open ball is a set of the form A subset of a metric space is open if and only if it is the union of open balls; it is closed if and only if its complement is open. A sequence converges to if and only if this is denoted by The closure If of a set consists of all points that are the limit of a sequence in; it is the smallest closed set containing. The interior A is the collection of all points such that for some open set G; it is the largest open set contained in. A function between two metric spaces is continuous at a point if and only if for every sequence it is continuous at everyif and only if the inverse image. A subset of a metric space is dense if and only if its closure is the whole space. A metric space is separable if and only if it has a countable dense subset. A subset of a metric space is compact if and only if it is closed and every sequence in has a converging subsequence.