F-door spaces and F-submaximal spaces

Abstract

[EN] Submaximal spaces and door spaces play an enigmatic role in topology. In this paper, reinforcing this role, we are concerned with reaching two main goals:

The first one is to characterize topological spaces X such that F(X) is a submaximal space (resp., door space) for some covariant functor Ff rom the category Top to itself. T0, and FH functors are completely studied.

Secondly, our interest is directed towards the characterization of maps f given by a flow (X, f) in the category Set, such that (X,P(f)) is submaximal (resp., door) where P(f) is a topology on X whose closed sets are exactly the f-invariant sets.