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)
Group operations among the open sets of a topological space.
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category-theory
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0It 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. – 2012-10-14
1 Answers
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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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2Well, no, no.. You stirred me up some other thoughts, but could not yet substantiate them.. Maybe you want to learn some about *locales*: http://en.wikipedia.org/wiki/Complete_Heyting_algebra – 2012-10-14