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pseudometric topology
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(Definition)
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Let be a pseudometric space. As in a metric space, we define
for ,
.
In the below, we show that the collection of sets
form a base for a topology for . We call this topology the pseudometric topology on induced by . Also, a topological space is a pseudometrizable topological space if there exists a pseudometric on whose pseudometric topology coincides with the given topology for [1,2].
Proposition 1
is a base for a topology.
Proof. We shall use the this result to prove that
 is a base.
First, as for all , it follows that
is a cover. Second, suppose
and
. We claim that there exists a
such that
By definition,
 and
 for some
 and
 . Then
Now we can define
 , and put
If  , then for  , we have by the triangle inequality
so
 and condition 1 holds. 
In the proof, we have not used the fact that is symmetric. 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.
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"pseudometric topology" is owned by matte.
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| Also defines: |
pseudometrizable, pseudometric topology, pseudo-metric, pseudometrizable topological space, pseudo-metrizable topological space |
This object's parent.
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Cross-references: induces, quasimetric, symmetric, proof, triangle inequality, cover, pseudometric, induced, topology, base, collection, metric space, pseudometric space
There are 3 references to this entry.
This is version 4 of pseudometric topology, born on 2004-10-03, modified 2007-06-02.
Object id is 6284, canonical name is PseudometricTopology.
Accessed 5797 times total.
Classification:
| AMS MSC: | 54E35 (General topology :: Spaces with richer structures :: Metric spaces, metrizability) |
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Pending Errata and Addenda
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