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Let $E$ be a topological space. For $x \in E$,the nhds of $x$ which are both closed and open form a fundamental system of nhds of $x$.Show that E is uniformizable. Check here for definition of uniformizable space. http://en.wikipedia.org/wiki/Uniformizable_space Hint given is the characteristic functions on such sets are continuous.This is a problem from Dieudonne

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HINT: Let $\Delta=\{\langle x,x\rangle\in X\times X:x\in X\}$, the diagonal in $X\times X$, and for each clopen set $H\subseteq X$ let $D_H=(H\times H)\cup\big((X\setminus H)\times(X\setminus H)\big)\;.$ Clearly each $D_H$ is an open nbhd of $\Delta$ in $X\times X$. Let $\mathscr{D}=\{D_H:H\text{ is a clopen subset of }X\}$, let $\mathscr{D}^*=\left\{\bigcap\mathscr{F}:\mathscr{F}\text{ is a finite subset of }\mathscr{D}\right\}\;,$ let $\mathscr{U}=\{U\subseteq X\times X:\exists D\in\mathscr{D}^*(D\subseteq U)\}$, and show that $\mathscr{U}$ is a diagonal uniformity on $X$ generating the original topology.

(In other words, $\mathscr{D}$ generates $\mathscr{U}$ in the same way that a subbase generates a topology.)

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    @K.Ghosh: You’re welcome.2012-12-20