Local Compactness in Hausdorff Space

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Theorem

Let $T = \left({S, \tau}\right)$ be a $T_2$ (Hausdorff) space.

Then in $T$ the following statements are equivalent:

$(1): \quad$ For each $x \in S$, there is a neighborhood base of compact subsets of $S$.
$(2): \quad$ Each $x \in S$ is contained in some compact neighborhood of $x$.

Proof

From the definition of locally compact, the first statement directly implies the second one.

For the converse implication, suppose that:

$\forall x \in S : \exists E^x \subseteq S$

where $E^x$ is a compact neighborhood of $x$.

Regard $E^x$ as a subspace of $T$.

every neighborhood of $x$ in $E^x$ is the intersection of $E^x$ and a neighborhood of $x$ in $T$.
every neighborhood of $x$ in $E^x$ is itself a neighborhood of $x$ in $T$.

Therefore:

every neighborhood basis of $x$ in $E^x$ is a neighborhood basis of $x$ in $T$.

As a consequence, it suffices to prove that every $E^x$ is locally compact.

From T2 Property is Hereditary, $E^x$ is a Hausdorff space.

By hypothesis, $E^x$ is compact.

Hence from Compact Hausdorff Space is Locally Compact, $E^x$ is locally compact.

$\blacksquare$