Every locally Compact Hausdorff Space is Regular.

Question

Every locally Compact Hausdorff Space is Regular

$Proof$:Let $X=$locally compact+$T_2\implies X^*$ is compact+$T_2\implies X^*$ is Normal($T_4$)$\implies X^*$ is Regular$\implies X$ is regular($T_3$)Subspace of regular space is regular)

Note:$X^*$ is one point compactification of $X$.

Are my Arguments true??

Answer 1

Yes that’s complete correct. Of course you do have to know (theorem for you?) that $X^\ast$ is indeed Hausdorff when $X$ is locally compact Hausdorff.