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Let $X$ be locally compact and Hausdorff. I want to prove the following:

If $Y\subset X$ is open then $Y$ is locally compact.

I have proved that closed subsets of $X$ are locally compact, but how can one prove this?

I also want to use these two lemmas to conclude the following:

If $Y\subset X$ is locally closed then $Y$ is locally compact.

2 Answers 2