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Let $(X,\mathscr T)$ be topological space. Prove that $ (X,\mathscr T)$ is $T_0$-space iff for each pair of $a$ and $b$ distinct members of X, $\overline{\{a\}}\neq \overline{\{b\}}.$

My attempt:-

Let $(X,\mathscr T)$ be topological space. Prove that $ (X,\mathscr T)$ is $T_0$-space. $a\neq b \implies$ there is an open set $U$ contains $a$ but not $b$. $\{b\}\subseteq X\setminus U \implies \overline{\{b\}}\subseteq X\setminus U.$ $a\notin X\setminus U$. Any closed set containing $a$ must be completeltely disjoint from $X\setminus U.$ So, $\overline{\{a\}}\neq \overline{\{b\}}.$

Conversaly, $\overline{\{a\}}\neq \overline{\{b\}}.$ Consider $X\setminus \overline{\{b\}}$, which is an open set. $X\setminus \overline{\{b\}} \subseteq X\setminus \{b\}$. $a\in X\setminus \{b\}$ . $b\notin X\setminus \{b\}\implies b\notin X\setminus \overline{\{b\}}$. How do I complete the proof?

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  • $\begingroup$ The first implication is not true as written: Give $\mathbb{R}$ the topology in which a proper subset $U\subset \mathbb{R}$ is open iff $0\not\in \mathbb{R}$, and put $a = 0$. You can swap $a$ and $b$, though. $\endgroup$ – anomaly Feb 11 at 15:42
  • $\begingroup$ Which implication? $\endgroup$ – Unknown x Feb 11 at 15:56
  • $\begingroup$ It is an excercise in the Foundation of topology by C.W Patty. $\endgroup$ – Unknown x Feb 11 at 15:59
  • $\begingroup$ See @PaulFrost's comment below. $\endgroup$ – anomaly Feb 11 at 16:21
  • $\begingroup$ You cannot prove both. That would be a $T_1$ space. Example: Sierpinski space : $S=\{c,d\}$ with $c\ne d$, and the only open sets are $S,\{c\}$, and $\emptyset. $ This is $T_0$ but not $T_1$. We have $ \overline {\{d\}}=\{d\}$ but $\overline {\{c\}}=\{c,d\}.$ $\endgroup$ – DanielWainfleet Feb 11 at 17:20
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The first part of your attempt starts with a wrong statement.

You say that if $X$ is $T_0$ and $a \ne b$, then there exists an open $U$ containing $a$ but not $b$. However, this is the definition of $T_1$. In a $T_0$ space you can only say there exists an open $U$ containing exactly one of $a, b$. Of course you can say that without loss of generality we may assume $a \in U$ and $b \notin U$. To see the difference, consider the Sierpinski space $\Sigma = \{ 0, 1 \}$ with topology $\mathfrak{T} = \{ \emptyset, \{ 1 \}, \Sigma \}$. It is $T_0$ because the only two distinct points are $0, 1$ and $\{ 1 \}$ is an open set containing exactly on of these points whereas there does not exist an open set containing $0$ but not $1$.

Moreover, you cannot conclude that any closed set containing $a$ must be completely disjoint from $X \setminus U$. But it is irrelevant since $a \notin \overline{\{b \}}$, hence $\overline{\{a \}} \ne \overline{\{b \}}$. As an example consider $\Sigma$ and take $a = 1, b = 0, U = \{1 \}$. You have $\overline{\{ a \}} = \Sigma$ and $\overline{\{ b \}} = \{ 0\}$. In fact $\overline{\{ b \}} \subset X \setminus U$ and $a \notin X \setminus U$, but $\overline{\{ a \}}$ is not completely disjoint from $X \setminus U$.

For the converse, let $a, b$ distinct points of $X$. Then $\overline{\{a \}} \ne \overline{\{b \}}$. The set $S = \overline{\{a \}} \cap \overline{\{b \}}$ is closed.

It is impossible that both $a, b \in S$ because otherwise $\overline{\{a \}} \subset S \subset \overline{\{b \}}$ and $\overline{\{b \}} \subset S \subset \overline{\{a \}}$, i.e. $\overline{\{a \}} = \overline{\{b \}}$.

Case 1. $S$ contains none of $a, b$. Then $b \notin \overline{\{a \}}$ (because otherwise $b \in \overline{\{a \}} \cap \overline{\{b \}} = S$), hence $b \in X \setminus \overline{\{a \}}$ and we are done since $a \in \overline{\{a \}}$, i.e. $a \notin X \setminus \overline{\{a \}}$.

Case 2. $S$ contains exactly one of $a, b$, w.l.o.g. $a$. Then $a \notin X \setminus S$ and $b \in X \setminus S$ and we are done again.

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  • $\begingroup$ But my converse is not complete. How do I prove that $a\in X\setminus \overline{ \{b\}}$? $\endgroup$ – Unknown x Feb 11 at 16:23
  • $\begingroup$ Thank you for correcting me in the first part. $\endgroup$ – Unknown x Feb 11 at 16:24
  • $\begingroup$ You have an amusing typo "unclean" for "unclear" in the 1st sentence. $\endgroup$ – DanielWainfleet Feb 11 at 17:07
  • $\begingroup$ @DanielWainfleet It was rather an unclear formulation of a non-native speaker than a typo ;-) I edit my answer. $\endgroup$ – Paul Frost Feb 12 at 10:53
  • $\begingroup$ My native language is English and I make a lot of typos here. $\endgroup$ – DanielWainfleet Feb 13 at 4:53

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