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Let X be a topological space and $\Omega \left(X \right)$ be the set of all open sets of X. Does anyone have a concrete example of X, where we can define a familiar algebraic structure on $\Omega\left(X\right)$, for example of a Group. Also, what is the categorical term for such things? (In contrast with group object in Top)

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$\Omega(X)$ is always a complete distributive lattice, with $\bigvee\mathscr{U}=\bigcup\mathscr{U}$ and $\bigwedge\mathscr{U}=\operatorname{int}\bigcap\mathscr{U}$ for $\mathscr{U}\subseteq\Omega(X)$. –  Brian M. Scott Oct 14 '12 at 15:49
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May I ask why you need this? –  Hui Yu Oct 14 '12 at 16:02
    
I am aware of that. I should have exclude that, since as you mentioned, the lattice is always there. For example, think of a finite Topology, whit the action of a dihedral group on its open sets. But, this is not a concrete example, since I am not suggesting any actual representation of open sets, as elements of a dihedral group. –  Hooman Oct 14 '12 at 16:07
    
It is possible to define a group structure on any set. It is just a matter of taste how well group structure and topology should fit together. –  Hagen von Eitzen Oct 14 '12 at 16:13
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Well.. it's not a usual approach of anything..

As Brian wrote in the first comment, $\Omega(X)$ becomes a complete distributive lattice naturally. Moreover, if $X$ is a top.group, then $\Omega(X)$ will also carry a semigroup structure.

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Yes. Thanks. I may delete this question then. –  Hooman Oct 14 '12 at 16:55
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Well, no, no.. You stirred me up some other thoughts, but could not yet substantiate them.. Maybe you want to learn some about locales: en.wikipedia.org/wiki/Complete_Heyting_algebra –  Berci Oct 14 '12 at 23:17
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