Equivalence of two topological conditions

Question

Are the following two conditions on a topological space $X$ equivalent?

1) $X$ is Hausdorff, second countable, and locally Euclidean. 2) $X$ is Hausdorff, second countable, and locally compact.

I know that 1) implies 2), what about the converse and is there any theorem about that?

Answer 1

Another counter example. A countable discete space.

Answer 2

Counterexample: in $\mathbb{R}^2=\{(x,y)|x,y\in \mathbb{R}\}$, consider $\{(x,0)\}\cup \{(0,y)\}$. It is Hausdorff, second countable, locally compact, but not locally Euclidean. To be specific, $(0,0)\in X$ has no open neighborhood isomorphic to Euclidean space.