Connected metric space, which is not locally compact.

Question

Are connected metric spaces locally compact too? If not, then I am looking for example of metric spaces which are connected, but not locally compact.

Answer 1

Let $X=\mathbb{R}^\omega$ be a countable product of copies of $\mathbb{R}$ in the product topology. This is a complete separable metric space that is (path-)connected (both properties are preserved by all products) but locally compact at no point: if $C$ were a compact neighbourhood of some point it would have to contain a basic open neighbourhood $U$ as well, and these have infinitely many coordinates $n$ such that $\pi_n[U]=\Bbb R$ which implies that $\pi_n[C] = \Bbb R$ for such $n$ as well, so $C$ cannot be compact.

Answer 2

Any infinite dimensional Banach space is connected (indeed path connected) but not locally compact.

Path connectivity follows from a simple check that $p(t)=t x + (1-t)y$ is a continuous path from $y$ to $x$. Lack of local compactness follows from the relatively well known theorem that a Banach space is locally compact if and only if it is finite dimensional.

Answer 3

The simplest counterexample I can find is $$\{(0,0)\} \cup \{(x,y) : x>0\}\subseteq \mathbb{R}^2$$

Here, we can see $(0,0)$ doesn't have a compact neighborhood as a result of no neighborhood of $(0,0)$ being complete.