Subspace of Euclidean space

Question

Consider the euclidean space $\mathbb{R}^n$.

Consider a closed compact subset $A\subset\mathbb{R}^n$. For example, take $A=[a,b]^n$, with $0<a<b<\infty$.

It is well know that $\mathbb{R}^n$ with appropriate euclidean distance is a locally compact separable metric space.

I know that every locally closed subspace of a locally compact space is locally compact.

Therefore, can we deduce from this that $[a,b]^n$ is locally compact and separable ?

I remind here the definition of separable space: a topological space is called separable if it contains a countable, dense subset.

Thank you for your help.

Answer 1

Any compact space is locally compact. $[a,b]^{n}$ is compact, hence locally compact.Any subspace of a seprable metric space is separable. Since $\mathbb R^{n}$ is separable so is $[a,b]^{n}$. In fact, points with rational coordinates form a countable dense subset of $[a,b]^{n}$.