202303021303

Type :Note Tags : Topology

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Local Compactness

Note

A space $X$ is called locally compact at $x$ if there is a compact subspace $C$ of $X$ such that it contains an open nbhd $V$ of $x$. $x\in V\subset C\subset X$

Proposition

Note

Let $X$ be a Hausdorff space. Then $X$ is locally compact iff for every $x\in X$ and every $U$ such that $x\in U\in {\tau }_X$, there is a neighbourhood $V$ of $x$ such that $x\in V\subset Cl(V)\subset U\subset X$ and $Cl(V)$ is compact.

Proof:

If $X$ is locally compact, then take $x\in X$ and $U$ a nbhd of $x$. Since $X$ is locally compact, we have a compact subset $C$ of $X$ (it is also closed since $X$ is hausdorff) which contains an open nbhd $V$ of $x$. Now take $W=V\cap U$ and $K=C-W$, note that $K$ is a closed subset of $C$ and hence is compact, and doesn’t contain $x$. Therefore, there is a pair of disjoint open sets $V_1,V_2$ s.t. $x\in V_1,K\subset V_2$. This gives $Cl(V_1\cap W)\subset W\subset U$ and we are done since $Cl(V_1\cap W)\subset C$ and hence is compact.

The other direction is trivial.

Proposition

Note

Let $X$ be locally compact Hausdorff. If $A\subset X$ is an open or closed subset of $X$, then $A$ is locally compact.

Proof:

1. If $A$ is closed: Since $X$ is locally compact, there is a compact set $C$ and a nbhd $U$ of $x$ such that $x\in U\subset C$. Intersected everything with A.
2. If A is open: Use the proposition above.

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Related Problems

1. $\mathbb{Q}$ is not locally compact.
2. Any compact space is locally compact.
3. ${\mathbb{R}}^n$ is locally compact.
4. ${\mathbb{R}}^{\mathbb{N}}$ is not locally compact in the product topology.
5. One Point Compactification

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References

Compactness Hausdorff Property