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.

Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results.

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