pseudometric topology

Let (X,d) be a pseudometric space. As in a metric space, we define


for xX, ε>0.

In the below, we show that the collection of sets


form a base for a topologyMathworldPlanetmath for X. We call this topology the on X induced by d. Also, a topological space X is a pseudometrizable topological space if there exists a pseudometric d on X whose pseudometric topology coincides with the given topology for X [1, 2].

Proposition 1.

is a base for a topology.


We shall use the result to prove that is a base.

First, as d(x,x)=0 for all xX, it follows that is a cover. Second, suppose B1,B2 and zB1B2. We claim that there exists a B3 such that

z B3B1B2. (1)

By definition, B1=Bε1(x1) and B2=Bε2(x2) for some x1,x2X and ε1,ε2>0. Then


Now we can define δ=min{ε1-d(x1,z),ε2-d(x2,z)}>0, and put


If yB3, then for k=1,2, we have by the triangle inequalityMathworldMathworldPlanetmath

d(xk,y) d(xk,z)+d(z,y)
< d(xk,z)+δ

so B3Bk and condition 1 holds. ∎


In the proof, we have not used the fact that d is symmetricMathworldPlanetmath. Therefore, we have, in fact, also shown that any quasimetric induces a topology.


  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
  • 2 S. Willard, General Topology, Addison-Wesley, Publishing Company, 1970.
Title pseudometric topology
Canonical name PseudometricTopology
Date of creation 2013-03-22 14:40:47
Last modified on 2013-03-22 14:40:47
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type Definition
Classification msc 54E35
Defines pseudometrizable
Defines pseudometric topology
Defines pseudo-metric
Defines pseudometrizable topological space
Defines pseudo-metrizable topological space