pseudometric topology
Let $(X,d)$ be a pseudometric space. As in a metric space, we define
$$ |
for $x\in X$, $\epsilon >0$.
In the below, we show that the collection of sets
$$\mathcal{B}=\{{B}_{\epsilon}(x)\mid \epsilon >0,x\in X\}$$ |
form a base for a topology^{} for $X$. We call this topology the on $X$ induced by $d$. Also, a topological space $X$ is a pseudometrizable topological space if there exists a pseudometric $d$ on $X$ whose pseudometric topology coincides with the given topology for $X$ [1, 2].
Proposition 1.
$\mathcal{B}$ is a base for a topology.
Proof.
We shall use the http://planetmath.org/node/5845this result to prove that $\mathcal{B}$ is a base.
First, as $d(x,x)=0$ for all $x\in X$, it follows that $\mathcal{B}$ is a cover. Second, suppose ${B}_{1},{B}_{2}\in \mathcal{B}$ and $z\in {B}_{1}\cap {B}_{2}$. We claim that there exists a ${B}_{3}\in \mathcal{B}$ such that
$z$ | $\in $ | ${B}_{3}\subseteq {B}_{1}\cap {B}_{2}.$ | (1) |
By definition, ${B}_{1}={B}_{{\epsilon}_{1}}({x}_{1})$ and ${B}_{2}={B}_{{\epsilon}_{2}}({x}_{2})$ for some ${x}_{1},{x}_{2}\in X$ and ${\epsilon}_{1},{\epsilon}_{2}>0$. Then
$$ |
Now we can define $\delta =\mathrm{min}\{{\epsilon}_{1}-d({x}_{1},z),{\epsilon}_{2}-d({x}_{2},z)\}>0$, and put
$${B}_{3}={B}_{\delta}(z).$$ |
If $y\in {B}_{3}$, then for $k=1,2$, we have by the triangle inequality^{}
$d({x}_{k},y)$ | $\le $ | $d({x}_{k},z)+d(z,y)$ | ||
$$ | $d({x}_{k},z)+\delta $ | |||
$\le $ | ${\epsilon}_{k},$ |
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 |
---|---|
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 |