Abelian torsion groups with a pseudocompact group topology

Two questions are posed: (a) Which Abelian torsion groups admit a PGT (pseudocompact group topology)? (b) If an Abelian torsion group G admits a PGT, for which cardinal numbers a may such a topology F be chosen so that the weight of the space > G,F > is equal to a? The authors answer question (a) completely (Theorems 3.17 and 3.19). In Theorem 3.24 for α > γ > ω they characterize those Abelian torsion groups of cardinality y which admit a PGT of weight a. This furnishes partial answers to (b)

On the relations P(X×Y)=PX×PY

AbstractLet P and Q be epireflective full subcategories of the category Haus of Hausdorff spaces and continuous functions, and also denote the corresponding reflectors by P: Haus → P and Q: Haus → Q respectively. Denote the class of P-regular spaces, i.e., of subspaces of P-spaces, by RP. Embracing certain special cases which have been treated in the literature, we show that if P ⊂ Q ⊂ RP then for X, Y ϵ RP the relation P(X × Y) = PX × PY implies Q(X × Y) = QX × QY. Applications to particular classes P,Q are given

Long chains of topological group topologies—A continuation

AbstractWe continue the work initiated in our earlier article (J. Pure Appl. Algebra 70 (1991) 53–72); as there, for G a group let B(G) (respectively N(G)) be the set of Hausdorff group topologies on G which are (respectively are not) totally bounded. In this abstract let A be the class of (discrete) maximally almost periodic groups G such that ¦G¦ = ¦GG′¦. We show (Theorem 3.3(A)) for G ϵ A with ¦G¦ = γ ⩾ ω that the condition that B(G) contains a chain C with ¦C¦ = β is equivalent to a natural and purely set-theoretic condition, namely that the partially ordered set 〈P(2γ), ⊆ 〉 contains a chain of length β. (Thus the algebraic structure of G is irrelevant.) Similar results hold for chains in B(G) of fixed local weight, and for chains in N(G).Theorem 6.4. If T1 ϵ B(G) and the Weil completion 〈(G,T1〉 is connected, then for every Hausdorff group topology T0 ⊆ T1 with ω〈G,T0〉 < α1 = ω〈G,T1〉 there are 2α1-many gro topologies between T0 and T1.From Theorem 7.4. Let F be a compact, connected Lie group with trivial center. Then the product topology T0 on Fω is the only pseudocompact group topology on Fω, but there are chains C ⊆ B(Fω) and C′ ⊆ B(Fω) with ¦C¦ = (2c+ and ¦C′¦ = 2(c+)such that T0 ⊆ ∩C and T0 ⊆ ∩C′

On the product of homogeneous spaces

AbstractWithin the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G0, G1 such that G0 × G1 is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenskiĭ, is this: In ZFC there are pseudocompact, homogeneous spaces X0, X1 such that X0 × X1 is not pseudocompact; if in addition MA is assumed, the spaces Xi may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α

Maximal independent families and a topological consequence

AbstractFor κ⩾ω and X a set, a family A⊆P(X) is said to be κ-independent on X if |⋂A∈FAf(A)|⩾κ for each F∈[A]<ω and f∈{−1,+1}F; here A+1=A and A−1=X⧹A.Theorem 3.6For κ⩾ω, some A⊆P(κ) with |A|=2κ is simultaneously maximal κ-independent and maximal ω-independent on κ. The family A may be chosen so that every two elements of κ are separated by 2κ-many elements of A.Corollary 5.4For κ⩾ω there is a dense subset D of {0,1}2κ such that each nonempty open U⊆D satisfies |U|=d(U)=κ and no subset of D is resolvable. The set D may be chosen so that every two of its elements differ in 2κ-many coordinates.Remarks(a) Theorem 3.6 answers affirmatively a question of Eckertson [Topology Appl. 79 (1997) 1–11]. Two proofs are given here. (b) Parts of Corollary 5.4 have been obtained by other methods by Feng [Topology Appl. 105 (2000) 31–36] and (for κ=ω) by Alas et al. [Topology Appl. 107 (2000) 259–273]

On the continuity of factorizations

[EN] Let {Xi : i ∈ I} be a set of sets, XJ :=Пi∈J Xi when Ø ≠ J ⊆ I; Y be a subset of XI , Z be a set, and f : Y → Z. Then f is said to depend on J if p, q ∈ Y , pJ = qJ ⇒ f(p) = f(q); in this case, fJ : πJ [Y ] → Z is well-defined by the rule f = fJ ◦ πJ|Y When the Xi and Z are spaces and f : Y → Z is continuous with Y dense in XI , several natural questions arise: (a) does f depend on some small J ⊆ I? (b) if it does, when is fJ continuous? (c) if fJ is continuous, when does it extend to continuous fJ : XJ → Z? (d) if fJ so extends, when does f extend to continuous f : XI → Z? (e) if f depends on some J ⊆ I and f extends to continuous f : XI → Z, when does f also depend on J? The authors offer answers (some complete, some partial) to some of these questions, together with relevant counterexamples. Theorem 1. f has a continuous extension f : XI → Z that depends on J if and only if fJ is continuous and has a continuous extension fJ : XJ → Z. Example 1. For ω ≤ k ≤ c there are a dense subset Y of [0, 1]k and f ∈ C(Y, [0, 1]) such that f depends on every nonempty J ⊆ k, there is no J ∈ [k]<ω such that fJ is continuous, and f extends continuously over [0, 1]k. Example 2. There are a Tychonoff space XI, dense Y ⊆ XI, f ∈ C(Y ), and J ∈ [I]<ω such that f depends on J, πJ [Y ] is C-embedded in XJ , and f does not extend continuously over XI .Comfort, W.; Gotchev, IS.; Recoder-Nuñez, L. (2008). On the continuity of factorizations. Applied General Topology. 9(2):263-280. doi:10.4995/agt.2008.1806.SWORD2632809

Isospin Mixing in the 1+ Doublet in 12-C

This work was supported by National Science Foundation Grant PHY 75-00289 and Indiana Universit