Metric Topology

A topology induced by the metric g defined on a metric space X. The open sets are all subsets that can be realized as the unions of open balls

 B(x_0,r)={x in X|g(x_0,x)<r},

where x_0 in X, and r>0.

The metric topology makes X a T2-space. Given two distinct points x_1 and x_2 of X, their distance r=g(x_1,x_2) is certainly positive, so the open balls B(x_1,r/2) and B(x_2,r/2) are disjoint neighborhoods of x_1 and x_2, respectively.

See also

Euclidean Topology, Metric Space, Metrizable Topology, Pseudometric Topology

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Metric Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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