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Let $T$ be an topology on $\mathbb{R}$ defined by $U\in T$ if and only if either $U$ doesn't contain $1$ or $U$ contains $0$. Would you help me how to check whether $T$ satisfiying separation axiom $T_0,T_1,T_2,T_3,T_4$. Thanks.

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    (I assume this is some sort of homework?) HINT: Many of those axoims deal with pairs of points. What's a natural pair of points to check them for?2012-12-24
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    By your definition, there are nontrivial open sets containing $1$. Also, think about which of these axioms imply which others.2012-12-24
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    Note that $0 \neq 1$. Let $V$ be any open set containing $1$, then by definition $0\in V$, hence $V$ always intersect open set that contain $0$. Hence, $T$ is not $T_2$. Since $T_3$ and $T_4$ implies $T_2$ then $T$ is not $T_3,T_4$2012-12-24

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