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Let $T_1=\{U:X-U \text{ is finite for all of } X\}$. Then $T_1$ is the cofinite topolgy on $X$,where $X$ is an arbitrary infinite set. Then $T_1$ is not a Hausdorff space.Is it a regular space or a normal space?

Let $T_2=\{U:X-U \text{ is countable or all of }X\}$,then $T_2$ is the co-countable topology.This also not a Hausdorff space.But the convergent sequences have unique limits.Is it true? Is $T_2$ a regular or normal space?

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    You might want to require that $X$ is uncountable for the second, otherwise you get the discrete topology which is Hausdorff. I would also avoid using $T_1$ and $T_2$ as the topologies as these often denote separation axioms -- which is incidentally the topic of the question!2012-12-17

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