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)

Thanks in advance!