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Let $X$ be a uniform space with uniformity $\mathcal{U}$ and natural topology of it is locally compact and paracompact space. (the natural topology, $\tau_{\mathcal{U}}$, on $X$ is the family of all sub sets $T$ of $X$ such that for every $x$ in $T$, there is $U\in\mathcal{U}$ such that $U[x]\subseteq T$.)

Since $(X, \tau_{\mathcal{U}})$ is locally compact hence for every $x\in X$ there is an open, relatively compact neighborhood $A_x$. Since $X$ is paracompact, the open cover $\{A_x\}_{x\in X}$ has a closed locally finite refinement $\{B_\alpha\}$. This implies that $M=\bigcup_{\alpha}(B_\alpha\times B_\alpha)$ is a proper, this means that for every compact subset $S$, the set $M[S]=\cup_{x\in S}M[x]$ is compact.

In my research, I must to work with entourage.

Q. Suppose that uniform space $(X, \mathcal{U})$ is a locally compact and paracompact space. Can I choose $V_{\alpha}\in\mathcal{U}$ such that $M=\cup_{\alpha}V_\alpha$ is proper?

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  • $\begingroup$ What is $M[x]$? $\endgroup$ – Paul Sinclair Feb 6 at 0:34
  • $\begingroup$ @PaulSinclair $M[x]=\{y: (x, y)\in M\}$ $\endgroup$ – user479859 Feb 6 at 4:28

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