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)
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.