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Pseudometric Topology


A topology on a set X whose open sets are the unions of open balls

 B(X_0,r)={x in x|g(x_0,x)<r},

where g is a pseudometric on X, x_0 is any point of X, and r>0.

There is a remarkable difference between a metric and a pseudometric topology. The former is always T_4, whereas the latter is, in general, not even T_0. In fact, a pseudometric allows g(x,y)=0 for some distinct points x and y, and then every open ball containing x contains y and conversely, so that no open set can separate the two points.


See also

Metric Topology

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Pseudometric Topology." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PseudometricTopology.html

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