Monads in topology

Let T be a submonad of the ultrafilter monad β and let G be a subfunctor of the filter functor. The T-algebras are topological spaces whose closed sets are the subalgebras and form thereby an equationally definable full subcategory of topological spaces. For appropriate T, countably generated free algebras provide ZFC examples of separable, Urysohn, countably compact, countably tight spaces which are neither compact nor sequential, and 2 c non-homeomorphic such examples exist. For any space X, say that U ⊂ X is G-open if U belongs to every ultrafilter in GX which converges in U. The full subcategory Top G consists of all G-spaces, those spaces in which every G-open set is open. Each Top G has at least these stability properties: it contains all Alexandroff spaces, and is closed under coproducts, quotients and locally closed subspaces. Examples include sequential spaces, P-spaces and countably tight spaces. T-algebras are characterized as the T-compact, T-Hausdorff T-spaces. Malyhinʹs theorem on countable tightness generalizes verbatim to Top G for any G ⊂ β . For r ∈ ω ⋆ = β ω \ ω , let G r be the subfunctor of β generated by r and let T r be the generated submonad. If ⩽ RK is the Rudin–Keisler preorder on ω ⋆ , r ⩽ RK s ⇔ G r ⊂ G s . Let ⩽ c be the Comfort preorder and define the monadic preorder r ⩽ m s to mean T r ⊂ T s . Then r ⩽ RK s ⇒ r ⩽ m s ⇒ r ⩽ c s . It follows that there exist 2 c monadic types. For each such type T r , the T r -algebras form an equationally definable full subcategory of topological spaces with only one operation of countably infinite arity. No two of these varieties are term equivalent nor is any one a full subcategory of another inside topological spaces. Say that r ∈ ω ⋆ is an m-point if G r ≠ T r . Under CH, m-points exist.