Difference between $T_1$ spaces and $T_2$ spaces

Question

I am really confused about the difference between $T_1$ space and $T_2$ space. To me they both seem to both have the same definition.

If $X=\{a,b,c,d\}$, what are the Topologies on $X$ that are $T_1$ but not $T_2$ and why?

Answer 1

There's no difference between $T_1$ and $T_2$ when considering finite topological spaces.

The reason is that a finite topological space has only finitely-many open sets. $T_1$ implies that a point $\{x\}$ is the intersection of all the open sets containing $x$. But since there are only finitely-many open sets, and finite intersections of open sets are open, it follows that $\{x\}$ is open. But then this means that the topology is discrete, in particular is $T_2$ (and much more).

This means we need to go to topological spaces with infinite underlying sets to find examples of spaces which are $T_1$ but not $T_2$. Such an example is the cofinite topology on any infinite set, say $\mathbb{N}$.