quality:correction
Generalizations of open sets

Throughout, (X, τ) {\displaystyle (X,\tau)} will be a topological space.

A subset A ⊆ X {\displaystyle A\subseteq X} of a topological space X {\displaystyle X} is called:

Using the fact that

whenever two subsets A, B ⊆ X {\displaystyle A,B\subseteq X} satisfy A ⊆ B, {\displaystyle A\subseteq B,} the following may be deduced:

Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set. Preopen sets need not be semi-open and semi-open sets need not be preopen.

Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space (X, τ) {\displaystyle (X,\tau)} forms a topology on X {\displaystyle X} that is finer than τ. {\displaystyle \tau.}

A topological space X {\displaystyle X} is Hausdorff if and only if every compact subspace of X {\displaystyle X} is θ-closed. A space X {\displaystyle X} is totally disconnected if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the closure of every preopen subset is open.