Topology, Hausdorff and Fréchet space

In case of finite set $X$, the cofinite topology and the discrete topology coincidies and hence each singleton is closed and open.

In case $X$ is infinite.

Then $X\setminus\{p\}$ is open hence $\{p\}$ is closed.

and for $p\neq a$.

$X\setminus (X\setminus\{a\})=\{a\}$ is finite and hence $X\setminus\{a\}$ is open and hence $\{a\}$ is closed.

Hence $X$ is Frechet.

Now for $x,y\in X$ and $x\neq y$ .

If one of $x$ or $y$ is $p$ say $x=p$.

Then $\{y\}$ is open And $X\setminus \{y\}$ is an open set such that $p\in X\setminus \{y\}$ and both are disjoint.

If $x\neq p$ and $y\neq p$ .

Then $\{x\}$ and $\{y\}$ are two open disjoint sets.

Then $X$ is Hausdorff.