## Compactification

Hi,

Let $X$ be non-compact topological space and define $X^{*}=X \cup \{\infty\}$ with the topology $\mathcal{T}^{*}=\{U \subset X^{*}| X \cap U \in \mathcal{T} \ \mbox{and} \ \infty \in U \Rightarrow X \setminus U \ \mbox{is compact and closed}\}$

How can I prove that $X^{*}$ is compact? I know the proof with covers and subcovers, but I was wondering how I can prove it only using filters (thus the fact that $X^{*}$ is compact if and only if every ultrafilter converges)