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[parent] pseudometric topology (Definition)

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

$\displaystyle 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

$\displaystyle \mathscr{B}= \{ B_\varepsilon(x)\mid \varepsilon>0, x\in X\} $
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   $ \mathscr{B}$ is a base for a topology.
Proof. We shall use the this result to prove that $ \mathscr{B}$ is a base.

First, as $ d(x,x)=0$ for all $ x\in X$, it follows that $ \mathscr{B}$ is a cover. Second, suppose $ B_1,B_2\in \mathscr{B}$ and $ z\in B_1\cap B_2$. We claim that there exists a $ B_3\in \mathscr{B}$ such that

$\displaystyle z$ $\displaystyle \in$ $\displaystyle B_3\subseteq B_1\cap B_2.$ (1)

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
$\displaystyle 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
$\displaystyle B_3 = B_\delta(z). $
If $ y\in B_3$, then for $ k=1,2$, we have by the triangle inequality
$\displaystyle d(x_k,y)$ $\displaystyle \le$ $\displaystyle d(x_k, z) + d(z,y)$  
  $\displaystyle <$ $\displaystyle d(x_k, z) + \delta$  
  $\displaystyle \le$ $\displaystyle \varepsilon_k,$  

so $ B_3\subseteq B_k$ and condition 1 holds. $ \qedsymbol$

Remark

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.

Bibliography

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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Also defines:  pseudometrizable, pseudometric topology, pseudo-metric, pseudometrizable topological space, pseudo-metrizable topological space

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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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