Is a bijective mapping from a compact set to a $T_1$ space is a homeomorphism? It is proved that the a bijective mapping from a compact set to a hausdorff space is a homeomorphism. Is the result true if hausdorffness is replaced by $T_1$ property? 

No, take any infinite compact Hausdorff space $X$ (e.g. $X=[0,1]$) and
$$X\to X$$
$$x\mapsto x$$
with the cofinite topology on the right side. It is clearly a continuous bijection. But it is not a homeomorphism because the right side is not Hausdorff. Indeed, the cofinite topology is Hausdorff only on finite spaces.
