Finding a topology on three point set not Hausdorff

Question

I have the set $X=\{4, 5, 6\}$ and want to find a topology on this that is not Hausdorff. I know that $T=\{\emptyset,X\}$ would work but I was looking for something more complex.

So trying to find a subset of $X$ that would work in a topology and I wondered if anyone knew what could work or my other thought was including the empty set within a subset but I don't know if this would work with $4, 5,6 $ included?

Hope that makes sense, happy to clarify anything

Thanks in advance!

Answer 1

Fix any point of $X$, call it $x_0$. Define your topology to have open sets (a) $\varnothing$ and (b) any subset that contains $x_0$. This is clearly a topology, and it cannot be Hausdorff since one can never find disjoint and non-empty open sets.

Answer 2

The topology $\tau=\{\emptyset,\{4\}, X\}$ works. Notice that $5$ and $6$ share all of their open sets (namely, just $X$). This space is not even $T_1$.

In fact, the only topology on a finite set that is Hausdorff is the discrete topology. See this question.