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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)

Mirko
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cr1t1cal
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  • Give $X$ the coarse topology, only $X$ and $\emptyset $ are open. – Cheerful Parsnip Dec 05 '15 at 17:58
  • 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 – cr1t1cal Dec 05 '15 at 18:03
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    Examples of countable connected Hausdorff spaces are given here: http://mathoverflow.net/questions/46986/countable-connected-hausdorff-space – Cheerful Parsnip Dec 05 '15 at 18:08
  • 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 – Mirko Dec 05 '15 at 18:32
  • For countable connected Hausdorff spaces see also here and the earlier question to which it’s linked. – Brian M. Scott Dec 05 '15 at 18:45
  • You might want to edit your question to rule out finite topologies, as well as the coarse topology. – Lee Mosher Dec 06 '15 at 00:14
  • For compact Hausdorff connected spaces we know that they contain a copy of the Cantor set (see http://mathoverflow.net/questions/38450/compact-hausdorff-spaces-without-isolated-points-in-zf) (connected is much stronger than having no isolated points) e.g. so there the cardinality couldn't be smaller than continuum. – Henno Brandsma Dec 06 '15 at 08:29

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