Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements:
BPI(X): Every proper filter of (X) can be extended to an ultrafilter.
UF(X): (X) has a free ultrafilter.
We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent:
(i) BPI(ω).
(ii) The Tychonoff product , where 2 is the discrete space 0,1, is compact.
(iii) The Tychonoff product is compact.
(iv) In a Boolean algebra...
We show in ZF that: (i) Every subcompact metrizable space is completely metrizable, and every completely metrizable space is countably subcompact. (ii) A metrizable space is countably compact if and only if it is countably subcompact relative to . (iii) For every metrizable space , the following are equivalent:
(a) is compact;
(b) for every open filter of , ;
(c) is subcompact relative to . We also show: (iv) The negation of each of the statements, (a) every countably subcompact metrizable...
We study in ZF and in the class of spaces the web of implications/ non-implications between the notions of pseudocompactness, light compactness, countable compactness and some of their ZFC equivalents.
In the realm of metric spaces we show in ZF that: (i) A metric space is compact if and only if it is countably compact and for every , every cover by open balls of radius has a countable subcover. (ii) Every second countable metric space has a countable base consisting of open balls if and only if the axiom of countable choice restricted to subsets of holds true. (iii) A countably compact metric space is separable if and only if it is second countable.
We show that AC is equivalent to the assertion that every compact completely regular topology can be extended to a compact Tychonoff topology.
We show: (i) The countable axiom of choice is equivalent to each one of the statements: (a) a pseudometric space is sequentially compact iff its metric reflection is sequentially compact, (b) a pseudometric space is complete iff its metric reflection is complete. (ii) The countable multiple choice axiom is equivalent to the statement: (a) a pseudometric space is Weierstrass-compact iff its metric reflection is Weierstrass-compact. (iii) The axiom of choice is equivalent to each one of the...
(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of to Y has size ≤ |(ω)|”.
(ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”.
(iii) Under P(ω) the following two statements are equivalent:
(a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter.
(b) Every Boolean algebra of...
In ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) the following statements are shown to be equivalent: (i) The axiom of dependent choice. (ii) Products of compact Hausdorff spaces are Baire. (iii) Products of pseudocompact spaces are Baire. (iv) Products of countably compact, regular spaces are Baire. (v) Products of regular-closed spaces are Baire. (vi) Products of Čech-complete spaces are Baire. (vii) Products of pseudo-complete spaces are Baire.
We show that the statement CCFC = “” is equivalent to the CMC and, the axiom of choice AC is equivalent to the statement CFE = “”. We also show that AC is equivalent to each of the assertions: “”, “” and “”.
In ZF, i.e., the Zermelo-Fraenkel set theory minus the Axiom of Choice AC, we investigate the relationship between the Tychonoff product , where 2 is 2 = 0,1 with the discrete topology, and the Stone space S(X) of the Boolean algebra of all subsets of X, where X = ω,ℝ. We also study the possible placement of well-known topological statements which concern the cited spaces in the hierarchy of weak choice principles.
In the framework of ZF (Zermelo-Fraenkel set theory without the Axiom of Choice) we provide topological and Boolean-algebraic characterizations of the statements " is countably compact" and " is compact"
We show that:
(1) It is provable in ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC) that every compact scattered T₂ topological space is zero-dimensional.
(2) If every countable union of countable sets of reals is countable, then a countable compact T₂ space is scattered iff it is metrizable.
(3) If the real line ℝ can be expressed as a well-ordered union of well-orderable sets, then every countable compact zero-dimensional T₂ space...
We investigate the role that weak forms of the axiom of choice play in countable Tychonoff products, as well as countable disjoint unions, of Loeb and selective metric spaces.
We show that given infinite sets and a function which is onto and -to-one for some , the preimage of any ultrafilter of under extends to an ultrafilter. We prove that the latter result is, in some sense, the best possible by constructing a permutation model with a set of atoms and a finite-to-one onto function such that for each free ultrafilter of its preimage under does not extend to an ultrafilter. In addition, we show that in there exists an ultrafilter compact pseudometric...
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