One- point compactification

Question

Let $X$ be a locally compact, non-compact Hausdorff space. Consider $Y=X\cup \{\infty\}$ as one-point compactification of $X$. Also, we assume that $K$ is compact in $X$ and $U$ an open set of $\infty$ in $Y$. What can say about relation between $X-K$ and $U$?. Can I say that $(X-K)\cap U\neq \emptyset$?

Answer 1

Since $\infty\in U$ and $U$ is open in $Y$, $X\setminus U$ must be compact in $X$ (by definition of the one-point compactification). Suppose that $(X\setminus K)\cap U=\varnothing$. Then, it follows that $X\setminus K\subseteq X\setminus U$. Therefore, $$X=(X\setminus K)\cup K\subseteq (X\setminus U)\cup K\subseteq X.$$ You can then express $X$ as a union of two compact sets $X\setminus U$ and $K$, contradicting the non-compactness assumption.