Topology counter example

Question

-We known that every subspace of T2 space is T2space.. but If we reverse it -if A is subspace of X and A was a T2 space need to be space X T2space also.? If the statement is wrong plz given example.

Answer 1

No, a single point $\{x\}$ in any space $X$ is Hausdorff, regardless of what separation axioms $X$ satisfies.

For a more sophisticated example you can take any Hausdorff space $A$ and add a point to it $X:=A\sqcup\{*\}$ with the following topology: $U\subseteq X$ is open if and only if $U=X$ or $U\subseteq A$ is open. Forget about Hausdorfness, the special $\{*\}$ is not even closed in $X$. But $A\subseteq X$ is Hausdorff.

The statement isn't even true if we assume that $X$ has all proper subsets Hausdorff, as the trivial topology on $\{1,2\}$ shows.