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 Quote by AdrianZ for example Zariski topology, How do we show that it is non-Hausdorff? I'm just interested to know how we could see if a space is Hausdorff or not.
What do you mean with the Zariski topology here?? Do you mean the topology consisting of all cofinite sets, or do you mean the topology associated with a commutative ring??

Let's say you mean the former, then we have an infinite set X and a topology

$$\mathcal{T}=\{U\subseteq X~\vert~X\setminus U~\text{is finite}\}\cup \{\emptyset\}$$

Take two arbitrary non-empty open sets U and V. Then $U\cap V$ is nonempty (check this). So the space is Hausdorff because there don't exist disjoint open sets!