Locally compact metric space - Mathematics Stack Exchange So any incomplete locally compact metric space is a counter-example to "only if" Moreover, as mentioned Tsemo Aristide's answer, any non-compact metric space, even a proper one, has the same topology as some improper metric space
Locally Closed Immersion - Mathematics Stack Exchange But this work in exactly the opposite direction then the problem we have here Does anybody see how the auther here conclude that $\Delta_X$ is locally closed immersion?
Exact meaning of every 2d manifold is locally conformal flat Note that local conformal flatness is a property of Riemannian manifold, so you need to specify a Riemannian metric Amazingly, every 2-dimensional Riemannian manifold is locally conformally flat - this is the theorem you are referring to
When is the union of two locally closed subsets locally closed? It isn't true in general that the union of two locally closed subsets (i e , subsets of the form open $\\cap$ closed) are locally closed, so is there some standard condition that guarantees it?