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Collection Proof Metric Space Continuation Of Exponent Subgraph Topological Space Basis Topological Space Symmetric Matrix Cover Triangle Inequality Pseudometric Space Quasimetric Space

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Pseudometric Topology

Let 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

. We call this topology the pseudometric topology on

induced by

. Also, a topological space

is a pseudometrizable topological space if there exists a pseudometric

whose pseudometric topology coincides with the given topology for

[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 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

, 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

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.

1	Topological Space
1	Metric Space
1	Proof
1	Collection
1	Subgraph
1	Continuation Of Exponent
2	Basis Topological Space
3	Symmetric Matrix
3	Cover
3	Triangle Inequality
6	Pseudometric Space
10	Quasimetric Space