**T** Let $X$ be $T_2$; $K\subseteq X$ compact. Then $K$ is closed.
**P** Let $K'=X\setminus K$. Pick $x\in K'$. Take for each $p\in K$ an open nbhd $N_p$ such that $x\notin N_p$. Then $K\subseteq \bigcup_{p\in K}N_p$. By compactness, $K\subset \bigcup_{i=1}^n N_{p_i}$ for $i=1,\dots,n$. By $T_2$ness, take an open nbhd $N_x$ such that $N_x\cap N_{p_i}=\varnothing$. Then $N_x\cap K=\varnothing$ and thus $N_x\subseteq K'$, so $K'$ is open and $K$ is closed.