- Thread starter
- #1
- Mar 10, 2012
- 835
Suppose $X$ is locally compact and Hausdorff. Given $x \in X$ and an open set $U$ containing $x$. Find a compact set $C \subseteq U$ which contains an open set $V$ such that $x \in V$.
I tried to construct an open subset $K$ of $U$ such that $\overline{K} \subseteq U$ and $x \in K$. Since if this happens then $C=\overline{K}$ and $V=K$ do the job. But I have not been able to. I found a proof over the internet using "one point compactification" but this concept is not discussed in my book..so there must be another way. Please help.
I tried to construct an open subset $K$ of $U$ such that $\overline{K} \subseteq U$ and $x \in K$. Since if this happens then $C=\overline{K}$ and $V=K$ do the job. But I have not been able to. I found a proof over the internet using "one point compactification" but this concept is not discussed in my book..so there must be another way. Please help.