Countable sums and products of metrizable spaces in ZF

Abstract

Key words Axiom of choice, weak axioms of choice, compact and Lindelöf metrizable spaces, sums and products of metrizable spaces. MSC (2000)03E25, 54A35, 54D30, 54D65, 54D70, 54E35, 54E45We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.In this paper we continue with the study of compact and Lindelöf metric spaces which we started in [12] and [13]. Besides the notation and terminology used in the above-mentioned papers we shall need to establish some more here. Let (X, T ) be a topological space. X is said to be metrizable iff there is a metric d on X such that the topology T d on X induced by d coincides with T .Form 9: Every infinite set has a countably infinite subset. Form 418: The disjoint union of countably many metrizable spaces is metrizable.i ∈ N} is a family of metric spaces each having the property P , then the (Tychonoff) product X = i∈N X i has the property Q.CPM le (P, Q): If {(X i , T i ) : i ∈ N} is a family of metrizable topological spaces each having the property P , then the (Tychonoff) product X = i∈N X i is metrizable and has the property Q.CSM(P, Q): If {(X i , d i ) : i ∈ N} is a disjoint family of metric spaces each having the property P , then the sum (disjoint topological union, see [19]) X = i∈N X i has the property Q.CSM le (P, Q): If {(X i , T i ) : i ∈ N} is a disjoint family of metrizable topological spaces each having the property P , then the sum X = i∈N X i is metrizable and has the property Q.CAC(ML) (resp. CAC(MC)): If {(X i , d i ) : i ∈ N} is a family of Lindelöf (resp. compact) metric spaces, then {X i : i ∈ N} has a choice function.Notice that CAC(ML) (resp. CAC(MC)) is equivalent to the proposition: Products of countably many Lindelöf (resp. compact) metric spaces are non-empty.CAC(M le L) (resp. CAC(M le C)) : If {(X i , T i ) : i ∈ N} is a family of Lindelöf (resp. compact) metrizable topological spaces, then {X i : i ∈ N} has a choice function.

Countable products and countable sums of compact metrizable spaces in the absence of the Axiom of Choice

“…The forms from our next two definitions will be called forms of type CPM le ( , ). They are defined in the spirit of [22]. Definition 1.11.…”

“…Let M, C, S, 2 be the following properties: M-to be a metrizable space; C-to be a compact space; S-to be a separable space; 2-to be a second-countable space. For properties P, Q, R, T ∈ {M, C, S, 2}, we define the following forms: There are differences between the notation in Definitions 1.11-1.13 and in [22]. For instance, if P, T are topological properties, then our CPM le (P, MT ) and CPM le (P, T ) from Definition 1.11 can be non-equivalent, while the form CPM le (P, T ) in [22] coincides with the form CPM le (P, MT ) from our Definition 1.11.…”

“…This article is devoted to the forms of type CPM le ( , ) and CSM le ( , ). Although, to a great extent, this work can be regarded as a continuation of [22], many new results are included in the forthcoming sections.…”

Countable compact Hausdorff spaces need not be metrizable in ZF

“…These statements concern topological sums and Tychonoff products of spaces sharing particular properties and have been studied in [6], [11], and [12] (without the requirement that the coordinate spaces are countable). Proof.…”

Weak axioms of choice for metric spaces

“…(i) First we point out that the statements M(Loeb,hLoeb) and M(sel,hsel) hold true in every permutation model, since any Loeb (or selective) metric space is well orderable in such a model; see the Introduction. To establish our independence result, we first recall the description of the permutation model we constructed in [10]. The set of atoms A = {A n : n ∈ ω}, where A n = {a n,x : x ∈ R} and A n is ordered like the reals by ≤ n .…”