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pseudometric topology
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(Definition)
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Let $(X,d)$ be a pseudometric space. As in a metric space, we define $$ B_\varepsilon(x)=\{ y\in X\mid d(x,y)<\varepsilon \}. $$ for $x\in X$ , $\varepsilon>0$ .
In the below, we show that the collection of sets
form a base for a topology for $X$ . We call this topology the pseudometric topology 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.
Proof. We shall use the this result to prove that
 is a base.
First, as $d(x,x)=0$ for all $x\in X$ , it follows that
is a cover. Second, suppose
and $z\in B_1\cap B_2$ . We claim that there exists a
such that \begin{eqnarray} \label{ii} z&\in& B_3\subseteq B_1\cap B_2. \end{eqnarray}By definition, $B_1 = B_{\varepsilon_1}(x_1)$ and $B_2 = B_{\varepsilon_2}(x_2)$ for some $x_1,x_2\in X$ and $\varepsilon_1,\varepsilon_2>0$ . Then $$ d(x_1, z)<\varepsilon_1, \quad d(x_2, z)<\varepsilon_2. $$ Now we can define $\delta = \min\{ \varepsilon_1-d(x_1, z), \varepsilon_2-d(x_2, z)\}>0$ , and put $$ B_3 = B_\delta(z). $$ If $y\in B_3$ , then for $k=1,2$ , we have by the triangle
inequality \begin{eqnarray*} d(x_k,y) &\le & d(x_k, z) + d(z,y) \\ &< & d(x_k, z) + \delta \\ &\le & \varepsilon_k, \end{eqnarray*}so $B_3\subseteq B_k$ and condition holds. 
In the proof, we have not used the fact that $d$ 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 7358 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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