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
Uniformizability of a space
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functional-analysis
1 Answers
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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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0by uniformizable,I want to use the definition that its topology can be given by a family of pseudometric. Of course they are equivalent but dieudonne has that defn. – 2012-12-20
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1@K.Ghosh: There are several ways to define a uniformity using pseudometrics. If Dieudonné’s is like Engelking’s, you can do something very similar to what I did above. For each clopen $H$ let $d_H$ be the pseudometric defined by $d_H(x,y)=|\chi_H(x)-\chi_H(y)|$. For each finite family $\mathscr{H}$ of clopen sets define $$d_{\mathscr{F}}(x,y)=\max_{H\in\mathscr{H}}d_H(x,y) \;.$$ The collection of these pseudometrics $d_{\mathscr{H}}$ defines a uniformity generating the original topology. – 2012-12-20
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0@K.Ghosh: There’s a typo in that last comment: it should be $d_{\mathscr{H}}(x,y)$ in the displayed formula. – 2012-12-20
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0,it's ok.now i got the correct idea thanks – 2012-12-20
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0@K.Ghosh: You’re welcome. – 2012-12-20