# Basic question on local compactness

Let $$X$$ be a topological space.

Let $$A \subset F \subset X$$.

Let $$F$$ be a subspace of $$X$$.

Let $$A$$ be a subspace of $$F$$.

Suppose that under this subspace topology inherited from $$F$$, $$A$$ forms a locally compact topological space.

Now let $$A$$ is a subspace of $$X$$.

Then how can I show that under the subspace topology inherited from $$X$$, $$A$$ is a locally compact topological space?

• The relative topology on $A$ inherited by the relative topology on $F$ is the same as the relative topology on $A$ inherited from $X$.
– Ruy
Oct 27, 2020 at 19:17

Because the topology that $$A$$ inherits from $$F$$ is the same topology that it inherits from $$X$$ (since the topology of $$F$$ is the one that it inherits from $$X$$). That's so because, for both topologies, a subset $$O$$ of $$A$$ is open if and only if $$O=O^\star\cap A$$, for some open subset $$O^\star$$ of $$X$$.