Is there a proper set in a locally compact, paracompact uniform space?

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?

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