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I will leave to Yemon Choi discussing the answer from Gelfand-Raikov-Shilov's book (Commutative Normed Rings, I suppose?), and restrict myself to more recent discussions on the matter... There is an …

Under the additional assumption of finite topological dimension pointed by YCor in the comments to the OP, the answer is yes, see e.g. Theorem 10, pp. 120 of the paper of K. Whittington, Local connect …

Let $X$ be a Hausdorff locally convex vector space. Recall (my reference is the book of H. Jarchow, Locally Convex Spaces. B.G. Teubner, 1981) that we say that $X$ is a semi-Montel space if every boun …

This is more like a long comment on the notion of smoothness than an actual answer, which has already been provided by Jochen Wengenroth. It tries to address the follow-up question the OP posted as a …

What you seem (to me) to be asking is under which conditions on a Lorentzian manifold its Alexandrov topology not even $T_0$. If that is the case, then it is easy to see that if $(M,g)$ is not chronol …

For Lorentzian manifolds, the conformal completion need not be compact. A typical example is the universal covering of the $d$-dimensional anti-de Sitter space-time (the maximally symmetric solution o …

Try the book of Peter T. Johnstone, "Stone Spaces" (Cambridge University Press, 1982). He works in the language of locales, which is unfortunately completely alien to me. Hope it helps.

It was not clear to me at first what your question has to do with bornologies, but now (EDITED) I see it. Any collection $\nu$ of subsets of $X$ (I assume that $X$ is nonvoid through the remainder of …