Find a locally compact space $X$ with a subspace $A$ that is NOT locally compact.

Question

I'd like to find a locally compact space $X$ with a subspace $A$ that is NOT locally compact.

As from here, I know that if $A$ is closed and $X$ is Hausdorff, then $A$ is locally compact.

Anyone have an example that satisfies the condition?

Answer 1

$$A=\mathbb Q\subset X=\mathbb R$$

Answer 2

Take Cantor space $\{0,1\}^\omega$ as $\mathbf{X}$ (which is compact), and then note that the subspace $$\mathbf{Y} := \{p \in \{0,1\}^\omega \mid |\{i \in \mathbb{N} \mid p(i)=0\}| = \infty\}$$ is homeomorphic to $\mathbb{N}^\mathbb{N}$, which is not locally compact.

Answer 3

The subset $$A=(\Bbb R_{>0}\times\Bbb R)\cup\{(0,0)\}$$ of $\Bbb R^2$ is not locally compact, since $\Bbb R^2$ is Hausdorff but $A$ is not open in its closure $\overline A=\Bbb R_{\ge0}×\Bbb R$