| Version current |
Version 3 |
| Let $(X,d)$ be a pseudometric space. As in a metric space, we define |
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 \}. |
B_\varepsilon(x)=\{ y\in X\mid d(x,y)<\varepsilon \}. |
| $$ |
$$ |
| for $x\in X$, $\varepsilon>0$. |
for $x\in X$, $\varepsilon>0$. |
| |
|
| In the below, we show that the collection of sets |
In the below, we show that the collection of sets |
| $$ |
$$ |
| \mathscr{B}= \{ B_\varepsilon(x)\mid \varepsilon>0, x\in X\} |
\mathscr{B}= \{ B_\varepsilon(x)\mid \varepsilon>0, x\in X\} |
| $$ |
$$ |
| form a base for a topology for $X$. We call this topology |
form a base for a topology for $X$. We call this topology |
| the \PMlinkescapetext{\emph{pseudometric topology}} on $X$ |
the \PMlinkescapetext{\emph{pseudometric topology}} on $X$ |
| induced by $d$. Also, |
induced by $d$. Also, |
| a topological space $X$ is a \emph{pseudometrizable topological space} |
a topological space $X$ is a \emph{pseudometrizable topological space} |
| if there exists a pseudometric $d$ on $X$ whose |
if there exists a pseudometric $d$ on $X$ whose |
| pseudometric topology coincides with the given topology |
pseudometric topology coincides with the given topology |
| for $X$ \cite{kelley, willard}. |
for $X$ \cite{kelley, willard}. |
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|
|
|
| \begin{prop} |
\begin{prop} |
| $\mathscr{B}$ is a base for a topology. |
$\mathscr{B}$ is a base for a topology. |
| \end{prop} |
\end{prop} |
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|
|
|
| \begin{proof} We shall use the \PMlinkid{this result}{5845} |
\begin{proof} We shall use the \PMlinkid{this result}{5845} |
| to prove that $\mathscr{B}$ is a base. |
to prove that $\mathscr{B}$ is a base. |
|
|
| First, as $d(x,x)=0$ for all $x\in X$, it follows |
First, as $d(x,x)=0$ for all $x\in X$, it follows |
| that $\mathscr{B}$ is a cover. |
that $\mathscr{B}$ is a cover. |
| Second, suppose $B_1,B_2\in \mathscr{B}$ and $z\in B_1\cap B_2$. |
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 |
We claim that there exists a $B_3\in \mathscr{B}$ such that |
| \begin{eqnarray} |
\begin{eqnarray} |
| \label{ii} |
\label{ii} |
| z&\in& B_3\subseteq B_1\cap B_2. |
z&\in& B_3\subseteq B_1\cap B_2. |
| \end{eqnarray} |
\end{eqnarray} |
| By definition, $B_1 = B_{\varepsilon_1}(x_1)$ |
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 $B_2 = B_{\varepsilon_2}(x_2)$ for some $x_1,x_2\in X$ |
| and $\varepsilon_1,\varepsilon_2>0$. Then |
and $\varepsilon_1,\varepsilon_2>0$. Then |
| $$ |
$$ |
| d(x_1, z)<\varepsilon_1, \quad d(x_2, z)<\varepsilon_2. |
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 |
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). |
B_3 = B_\delta(z). |
| $$ |
$$ |
| If $y\in B_3$, then for $k=1,2$, we have by the triangle inequality |
If $y\in B_3$, then for $k=1,2$, we have by the triangle inequality |
| \begin{eqnarray*} |
\begin{eqnarray*} |
| d(x_k,y) &\le & d(x_k, z) + d(z,y) \\ |
d(x_k,y) &\le & d(x_k, z) + d(z,y) \\ |
| &< & d(x_k, z) + \delta \\ |
&< & d(x_k, z) + \delta \\ |
| &\le & \varepsilon_k, |
&\le & \varepsilon_k, |
| \end{eqnarray*} |
\end{eqnarray*} |
| so $B_3\subseteq B_k$ and condition \ref{ii} holds. |
so $B_3\subseteq B_k$ and condition \ref{ii} holds. |
| \end{proof} |
\end{proof} |
|
|
| \subsubsection*{Remark} |
\subsubsection*{Remark} |
| In the proof, we have not used the fact that $d$ is |
In the proof, we have not used the fact that $d$ is |
| symmetric. Therefore, we have, in fact, also shown that any |
symmetric. Therefore, we have, in fact, also shown that any |
| quasimetric induces a topology. |
quasimetric induces a topology. |
| |
|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{kelley} J.L. Kelley, \emph{General Topology}, |
\bibitem{kelley} J.L. Kelley, \emph{General Topology}, |
| D. van Nostrand Company, Inc., 1955. |
D. van Nostrand Company, Inc., 1955. |
| \bibitem{willard} S. Willard, \emph{General Topology}, |
\bibitem{willard} S. Willard, \emph{General Topology}, |
|
Addison-Wesley, Publishing Company, 1970.
|
Addison-Wesley, Publishing Company, 1968.
|
| \end{thebibliography} |
\end{thebibliography} |
