# Does there exist a $\sigma$-compact complete metric space which is not locally compact? [closed]

Give an example of a $$\sigma$$-compact complete metric space which is not locally compact. A space $$X$$ is said to be locally compact if for each $$x\in X$$, there exist an open set $$U$$ and a compact subset $$K$$ of $$X$$ such that $$x\in U\subseteq K$$.

Let $$\mathcal H$$ be a separable Hilbert space with basis $$v_1,v_2,\ldots$$ and let $$X$$ be the closed subspace of all vectors that can be written as $$tv_n$$ with $$t\in[0,1]$$ and $$n\in\mathbb N$$. Then $$X$$ is $$\sigma$$-compact, since it is the union of the compact sets $$K_n = \{tv_n \mid t\in [0,1]\}$$. But $$X$$ is not locally compact, because there is no compact neighbourhood of 0.
• I believe your space with an hedgehog space of spininess $\aleph_0$. Dec 24, 2021 at 12:23