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# 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 *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

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

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.

## Mathematics Subject Classification

54E35*no label found*

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