Decompositions of topological spaces and total recurrence
by
Oleksandr Petrenko
Kyiv National University, Department of Cybernetics
Coauthors: I.V. Protasov

We say that a subspace A of a topological space X is almost discrete if, for every a ∈ A, there exists a neighbourhood U of a in X such that A∩U is finite. If X is a T₁-space then every almost discrete subspace of X is discrete.

We say that an infinite topological space X is cofinite if each proper closed subspace of X is finite. Given an infinite set X, there exists only one cofinite T₁-topology t on X: U ∈ t if and only if U=∅ or X\U is finite.

Theorem 1 Every topological space X can be partitioned X=F∪AD∪CF where F is finite, AD is a disjoint union of countable almost discrete subspaces, CF is a disjoint union of cofinite subspaces.

We say that a subspace A of a topological space X is Hausdorff if, for any a, b ∈ A, there exist the neighbourhoods U, V of a, b in X such that U∩V = ∅

We say that a subspace A of a topological space X is linked if, for any a, b ∈ A and any neighbourhoods U, V of a, b in X one has U∩V ≠ ∅.

Theorem 2 Every topological space X can be partitioned X=F∪H∪L where F is finite, H is a disjoint union of infinite Hausdorff subspaces, L is a disjoint union of infinite linked subspaces.

By [2], every infinite Hausdorff space is either a one-point compactification of an infinite discrete space, or can be partitioned in countable discrete subspaces.

Theorem 3 Let X be an infinite topological space without cofinite subspaces. Then one of the following statements holds

(i) X is a disjoint union of countable almost discrete subspaces;
(ii) there exists a non-empty finite subspace F ⊂ X such that, for every a ∈ F and every neighbourhood U of a, X\U is finite.

A topological space X is called totally recurrent [1](resp. bijectively recurrent [2]) if every, not necessarily continuous, mapping (resp. bijection) f:X→X has a recurrent point. Recall that x ∈ X is recurrent if x is a limit point of the orbit (fⁿ(x))_{n ∈ w}. By [2], an infinite Hausdorff space X is totally (bijectively) recurrent if and only if X is a one-point compactification of an infinite discrete space.

Theorem 4 Let X be an infinite topological space without cofinite subspaces. Then one of the following statements are equivalent

(i) X is totally recurrent;
(ii) X is bijectively recurrent;
(iii) there exists a non-empty finite subspace F ⊂ X such that, for every a ∈ F and every neighbourhood U of a, X\U is finite.

Theorem 5 If a topological space X has no infinite almost discrete subspaces (in particular, X is cofinite), then X is totally recurrent.

[1] O.Petrenko, I.V.Protasov, Discontinuous Birkhoff Theorem, Ukr.Math.Bull, 4(2007), 429-431.
[2] O.Petrenko, I.V.Protasov, Around Birkhoff Theorem, Travaux Math. (to appear)

Date received: April 28, 2008