Versions of Normality and Some Weak Forms of the Axiom of Choice

Bull. Polish Acad. Sci. Math.

2017

Self Cite

Presented by Czesław BESSAGA Summary. Let X = (X, d) and Y = (Y, ρ) be two metric spaces. (a) We show in ZF that: (i) If X is separable and f : X → Y is a continuous function then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ(f (A), f (B)) = 0. But it is relatively consistent with ZF that there exist metric spaces X, Y and a continuous, nonuniformly continuous function f : X → Y such that for any A, B ⊆ X with d(A, B) = 0, ρ(f (A), f (B)) = 0. (ii) If S is a dense subset of X, Y is Cantor complete and f : S → Y a uniformly continuous function, then there is a unique uniformly continuous function F : X → Y extending f. But it is relatively consistent with ZF that there exist a metric space X, a complete metric space Y, a dense subset S of X and a uniformly continuous function f : S → Y that does not extend to a uniformly continuous function on X. (iii) X is complete iff for any Cauchy sequences (xn) n∈N and (yn) n∈N in X, if {xn : n ∈ N} ∩ {yn : n ∈ N} = ∅ then d({xn : n ∈ N}, {yn : n ∈ N}) > 0. (b) We show in ZF+CAC that if f : X → Y is a continuous function, then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ(f (A), f (B)) = 0.

“…Hence, by the Tietze extension theorem which holds in ZF (see e.g. [3,Lemma 2]), h extends to a continuous real valued g on X. By our hypothesis, ρ(g(A), g(B)) = 0.…”

mentioning

confidence: 79%

Uniform continuity and normality of metric spaces in $\mathbf{ZF}$

Bull. Polish Acad. Sci. Math.

2017

Self Cite

“…Of course, every T space is a U space and all U spaces are normal. In [16], NU (NT, resp.) is an abbreviation to: "Every normal space is a U space."…”

Section: Strange Hausdorff Compactificationsmentioning

confidence: 99%

“…Hence, in ZF + DC, a topological space X satisfies UL(X) if and only if TET(X) holds. In [16], it was shown that there is a model M of ZF in which there is a compact Hausdorff space X such that UL(X) holds and TET(X) fails in M. However, it is an open question, already posed in [16], whether UL implies TET.…”

Section: Strange Hausdorff Compactificationsmentioning

confidence: 99%

“…It was shown in [16] that vDCP(ω) is strictly weaker than CMC. For significant applications of models in which vDCP(ω) fails, the following construction was used, for instance, in [4], [16], [27] and in Section 4.7 of [12]:…”

Section: Proposition 212 [Zf]mentioning

confidence: 99%

“…(ii) Now, let us suppose that CMC is false. In this case, notice that it was shown in the proof to Theorem 3 of [16] that there exists a Tychonoff space Z such that, for a compact subset C of Z, the set Z \ C is dense in Z, while C is not C * -embedded in Z and C is homeomophic with ω + 1. Assume that UFT is satisfied.…”

Section: Proposition 212 [Zf]mentioning

confidence: 99%

See 2 more Smart Citations

Hausdorff compactifications in ZF

Topology and its Applications

Wajch

2019

Self Cite

For a compactification αX of a Tychonoff space X, the algebra of all functions f ∈ C(X) that are continuously extendable over αX is denoted by C α (X). It is shown that, in a model of ZF, it may happen that a discrete space X can have non-equivalent Hausdorff compactifications αX and γX such that C α (X) = C γ (X). Amorphous sets are applied to a proof that Glicksberg's theorem that βX × βY is the Čech-Stone compactification of X × Y when X × Y is a Tychonoff pseudocompact space is false in some models of ZF. It is noticed that if all Tychonoff compactifications of locally compact spaces had C *embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space X to generate a compactification of X are given in ZF. A concept of a functional Čech-Stone compactification is investigated in the absence of the axiom of choice.

Disjoint Unions of Topological Spaces and Choice

Howard

Mathematical Logic Qtrly

Rubin

et al. 1998

in the absence of some form of the Axiom of Choice.Mathematics Subject Classification: 03E25, 04A25, 54D10, 54D15.We find properties of topological spaces which are not shared by disjoint unions Introduction and terminologyThis is a continuation of the study of the roll the Axiom of Choice plays in general topology. See also [l], [2], [3], and [4]. Our primary concern will be the use of the axiom of choice in proving properties of disjoint unions of topological spaces (see Definition 1, part 11). For example, in set theory with choice the disjoint union of metrizable topological spaces is a metrizable topological space. The usual proof of this fact begins with the choice of metrics for the component spaces. We will show that the use of some form of choice cannot be avoided in this proof and in fact without choice the disjoint union of metrizable spaces may not even be metacompact.In Section 1 we show that many assertions about disjoint unions of topological spaces are equivalent to the axiom of multiple choice. Models of set theory and corresponding independence results are described in Section 2. In Section 3, we study the roll the Axiomof Choice plays in the properties of disjoint unions of collectionwise Hausdorff and collectionwise normal spaces.We begin with the definitions of the symbols and terms we will be using ')