# pseudometric topology

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

 $\mathscr{B}=\{B_{\varepsilon}(x)\mid\varepsilon>0,x\in X\}$

form a base for a topology 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.

$\mathscr{B}$ is a base for a topology.

###### Proof.

We shall use the http://planetmath.org/node/5845this 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

 $d(x_{1},z)<\varepsilon_{1},\quad d(x_{2},z)<\varepsilon_{2}.$

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

 $\displaystyle d(x_{k},y)$ $\displaystyle\leq$ $\displaystyle d(x_{k},z)+d(z,y)$ $\displaystyle<$ $\displaystyle d(x_{k},z)+\delta$ $\displaystyle\leq$ $\displaystyle\varepsilon_{k},$

so $B_{3}\subseteq B_{k}$ and condition 1 holds. ∎

## 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.

## References

• 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 PseudometricTopology 2013-03-22 14:40:47 2013-03-22 14:40:47 matte (1858) matte (1858) 7 matte (1858) Definition msc 54E35 pseudometrizable pseudometric topology pseudo-metric pseudometrizable topological space pseudo-metrizable topological space