F-door spaces and F-submaximal spaces

Abstract

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 F from the category Top to itself. T0, ρ 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.

F-nodec spaces

“…[5, Remarks 2.3] Let X be a topological space and A be a subset of X. In [5] the following remarks are given.…”

“…In [5], L. Dridi et al characterized topological spaces X such that F(X) is a submaximal space, for a given covariant functor F.…”

F-resolvable spaces

“…Now, as in the first section, we need to recall some notations and results introduced by the authors in [6].…”

“…, where H is a collection of continuous maps from X to R. Now, in order to characterize ρ-resolvable spaces and FH-resolvable spaces, we need to recall the following definition introduced in [6]. Theorem 3.6.…”

“…Furthermore, in [3] and [4] ( [6] and [7]), the authors give a characterization of spaces such that their F-reflections (their compactifications) are submaximal, door and nodec. In this paper F designates a covariant functor from the category Top to itself.…”