Metric Space - Open and Closed Sets, Topology and Convergence

Open and Closed Sets, Topology and Convergence

Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.

About any point in a metric space we define the open ball of radius about as the set

These open balls form the base for a topology on M, making it a topological space.

Explicitly, a subset of is called open if for every in there exists an such that is contained in . The complement of an open set is called closed. A neighborhood of the point is any subset of that contains an open ball about as a subset.

A topological space which can arise in this way from a metric space is called a metrizable space; see the article on metrization theorems for further details.

A sequence in a metric space is said to converge to the limit iff for every, there exists a natural number N such that for all . Equivalently, one can use the general definition of convergence available in all topological spaces.

A subset of the metric space is closed iff every sequence in that converges to a limit in has its limit in .