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for a topology $(X,T)$ of arbitrary sets . whats an example where $\forall a \neq b \in X $ where $a$ is an open set $U $ and $b \not \in U$ and there is not subset where V where $b \in V$ but $a \not \in V$

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was able to make an example fora finite way but could not generilized

example

$X= \{ a,b\}$ and let $T_0= \{\emptyset,X, \{ a \} \}$. So, $b \in X$ and $a \in X$ theres no subset of $T_0$ that contains $b$ but does not contain $a$

THought of the euclidean topology. Dont think that doesnt work. are integers a topology? Anyways, I think the answer has to do with discrete metric.

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    I think I understand what you are asking, but it's difficult to read the question as it is now. Are you asking the following? Give an example of a topological space $(X,T)$ such that for all $a,b\in X$ 1) there exists an open set $U\subseteq X$ such that $a\in U$, $b\notin U$ 2) for all proper open subsets $V\subsetneq X$ such that $b\in V$, we have $a\notin V$ – 2017-02-06
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    @john yeah not sure if it has to be a proper subset. . found an example http://mathonline.wikidot.com/hausdorff-topological-spaces-examples-3. . I think its its example 1 works. if a metric space is a topology – 2017-02-06
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    Why deleting the question? I think you should answer your own question with the example you found so that it can be helpful for the community! – 2017-02-06
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    ok, wont delete it. – 2017-02-06

1 Answers 1

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The only example is $X=\{a,b\}$ with $a\ne b$ and with every subset of $X$ being open.

(1).If there exist three distinct $a,b,c \in X $ then there exist open $U_a$ and open $U_b$ with $$a\in U_a\land c\not \in U_a\land b\in U_b\land c\not \in U_b,$$ implying that $U_a\cup U_b$ is an open set containing $a$ and $b$ but not $c.$

(2).If $X=\{a,b\}$ then the only subset of $X$ that contains $a$ but not $b$ is $\{a\}$, and the only subset of $X$ that contains $b$ but not $a$ is $\{b\}.$