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Say you are given a set X, does there always exist a non-trivial (not the coarse topology) topology on X such that X is

1) Comapact 2) Connected 3) Compact and Connected

Additionally, if it were possible can you give an example of such a topology on the set of Integers(where in the set of integers would be compact,connected)

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  • $\begingroup$ Give $X$ the coarse topology, only $X$ and $\emptyset $ are open. $\endgroup$ Dec 5, 2015 at 17:58
  • $\begingroup$ I'm sorry, I should have mentioned I was looking for a non trivial topology.I'll edit the question to a add that phrase $\endgroup$
    – cr1t1cal
    Dec 5, 2015 at 18:03
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    $\begingroup$ Examples of countable connected Hausdorff spaces are given here: mathoverflow.net/questions/46986/… $\endgroup$ Dec 5, 2015 at 18:08
  • $\begingroup$ The cofinite topology on a countable set is compact, connected and $T_1$ but not Hausdorff. The answer to (1) is trivial yes, given any (infinite) cardinal consider the Alexandroff compactification of a discrete space of that cardinality:The result is compact Hausdorff (hence normal) but not connected. If $X$ is finite and $T_1$ then is is compact but not connected (only connected if $X$ is a singleton). There are finite $T_0$ spaces that are connected, with applications in Digital topology $\endgroup$
    – Mirko
    Dec 5, 2015 at 18:32
  • $\begingroup$ For countable connected Hausdorff spaces see also here and the earlier question to which it’s linked. $\endgroup$ Dec 5, 2015 at 18:45

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