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I want to find all topologies $\tau$ on finite set with $n$ elements such that all topologies have exactly one open single point or exactly $\{a_0 \}$ be in topology and cardinal of topologies $\tau$ be $2^{n-1} <|\tau |$.

Topologies on a finite set $X$ are in one-to-one correspondence with preorders on $X$.

I find one topology such that exactly has one open single point set:

$$\tau=\{U:\{a_0\}\subseteq U\}\cup\{\varnothing\}. $$

Which topologies admit exactly one open singleton and have more than half of the possible subsets open? How can I find all such topologies?

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    $\begingroup$ This paper characterizes all topologies on an n-element set with $2^{n-1} < |\tau|$, at least implicitly. $\endgroup$
    – Ulli
    Commented Jun 16 at 20:14
  • $\begingroup$ Let $Y = X\setminus \{a_0\}$. For each of your topologies $\tau$, the subspace topology $\tau_Y$ on $Y$ has no single point open sets. For a given $\tau_Y$, all the topologies on $\tau$ that induce it can be constructed by first including the unions of each set in $\tau_Y$ with $\{a_0\}$, Then choose which sets in $\tau_Y$ you want to also be in $\tau$ directly, with the two provisios that $\emptyset$ must be included, and if $U \in \tau_Y$ is in $\tau$ then so must be every open subset of $U$ in $Y$. $\endgroup$
    – Paul Sinclair
    Commented Jun 17 at 19:11
  • $\begingroup$ Thus you can start with the topologies of $Y$ with no discrete points and more than $2^{n-2}$ open sets, then determine how many ways they can be completed to form a desired topology on $X$. $\endgroup$
    – Paul Sinclair
    Commented Jun 17 at 19:12

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As I said when you posted this question on Mathoverflow, I would still like to hear your motivation for asking this question, and I don't think it should receive a full answer until you've given one. That's why the question has a negative score on the other site.

It would also be good to hear what approaches you've tried toward investigating the problem more systematically. If you're completely stuck, Paul's comments about separately considering those open sets which do and don't contain $a_0$ are a good hint to start you off.

If that doesn't get you anywhere either, I'll consider posting a full solution to the problem if you first share your motivation for asking about it.

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  • $\begingroup$ This is a question from my thesis . I want to count all topologies with above conditions. $\endgroup$
    – amir bahadory
    Commented Jun 18 at 9:21
  • $\begingroup$ Also I need classify all these topologies $\endgroup$
    – amir bahadory
    Commented Jun 18 at 13:48
  • $\begingroup$ When I said "motivation", I meant mathematical motivation. Why in your thesis do you want to classify topologies with those conditions? Is it simply an exercise set by your supervisor, or are there more specific reasons why you want to consider this particular problem? Have you discussed it with your supervisor, and if so what have they suggested? Have you tried considering separately the open sets which do and don't contain $a_0$, and if so where did that lead you? $\endgroup$
    – Robin Saunders
    Commented Jun 19 at 11:13
  • $\begingroup$ Motivation: we want to solve problem about classify all topologies on finite set but for special case and we want generalize this case to another cases step by step. $\endgroup$
    – amir bahadory
    Commented Jun 28 at 17:18
  • $\begingroup$ So if I've understood correctly, your supervisor has asked you to work on classifying all topologies on $n$ points with exactly one open single point $a_0$, and to begin by classifying those where the topology is large ($|\tau| > 2^{n-1}$)? Is that the situation? Apart from the example you've identified, what have you tried so far? Have you considered separately the open sets which do and don't contain $a_0$, as Paul and I suggested? Where does that lead you? $\endgroup$
    – Robin Saunders
    Commented Jun 30 at 6:40

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