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For first-countable regular spaces (which form an important subclass of LCS-spaces) there exists a simple characterization: a first-countable regular space $X$ is not locally countably compact if and …

It seems that Gutik's hedgehog is a required counterexample. I recall that Gutik's hedgehog is the set $$X=\{(0,0)\}\cup\{(\tfrac1n,0):n\in\mathbb N\}\cup\{(\tfrac1n,\tfrac1{nm}):n,m\in\mathbb N\}$$ e …

This is not an answer to Muranov's problem. Below I present some partial results which can be interesting or helpful for researchers that will try to attack this problem in future. The proofs of the f …

It seems that in Banach spaces weakly compact subsets remain compact in the weak sequential topology. This follows from three facts: By a well-known Eberlein Theorem, weakly compact sets in Banach s …

In the "positive" direction, there exist some deep results (called Z-set unknotting theorems) on extensions of homeomorphisms between Z-sets of Menger manifolds, see Theorem 4.1.18 in the book "Inver …

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$. Question. Is there a non-compact linear bounded operato …