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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?

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

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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 $X$.

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