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1.
We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G?(K?A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups. 相似文献
2.
Within 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. Uspenski?, 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. 相似文献
3.
Aleena Sokolovskaya 《Topology and its Applications》2008,155(4):342-346
Examples of a pseudocompact (even countably compact) G-space which is not G-Tychonoff and of a locally compact pseudocompact (even countably compact) G-Tychonoff space X with βGX≠βX are constructed. 相似文献
4.
We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ).Every pseudocompact Abelian group G with cardinality |G|≤22c satisfies this inequality and therefore admits a pseudocompact group topology with property . Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property .We also observe that pseudocompact Abelian groups with property contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact. 相似文献
5.
Mikhail Tkachenko 《Topology and its Applications》2009,156(12):2158-2165
A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G∧ of G onto the dual group G∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ. 相似文献
6.
A completely regular space X is called nearly pseudocompact if υX?X is dense in βX?X, where βX is the Stone-?ech compactification of X and υX is its Hewitt realcompactification. After characterizing nearly pseudocompact spaces in a variety of ways, we show that X is nearly pseudocompact if it has a dense locally compact pseudocompact subspace, or if no point of X has a closed realcompact neighborhood. Moreover, every nearly pseudocompact space X is the union of two regular closed subsets X1, X2 such that Int X1 is locally compact, no points of X2 has a closed realcompact neighborhood, and . It follows that a product of two nearly pseudocompact spaces, one of which is locally compact, is also nearly pseudocompact. 相似文献
7.
Haruto Ohta 《Topology and its Applications》1984,17(3):265-274
In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω1, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact. 相似文献
8.
We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P, the class P∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment there is Y⊂X with Y∈P and . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments. 相似文献
9.
Tsugunori Nogura 《Topology and its Applications》1983,15(1):65-69
We study the question whether a topological space X with a property can be embedded in a countably compact space X? with the same property . 相似文献
10.
P is the class of pseudocompact Hausdorff topological groups, and P′ is the class of groups which admit a topology T such that (G,T)∈P. It is known that every G=(G,T)∈P is totally bounded, so for G∈P′ the supremum T∨(G) of all pseudocompact group topologies on G and the supremum T#(G) of all totally bounded group topologies on G satisfy T∨⊆T#.The authors conjecture for abelian G∈P′ that T∨=T#. That equality is established here for abelian G∈P′ with any of these (overlapping) properties. (a) G is a torsion group; (b) |G|?c2; (c) r0(G)=|G|=ω|G|; (d) |G| is a strong limit cardinal, and r0(G)=|G|; (e) some topology T with (G,T)∈P satisfies w(G,T)?c; (f) some pseudocompact group topology on G is metrizable; (g) G admits a compact group topology, and r0(G)=|G|. Furthermore, the product of finitely many abelian G∈P′, each with the property T∨(G)=T#(G), has the same property. 相似文献
11.
Dmitri Shakhmatov 《Topology and its Applications》2009,156(7):1216-1223
If a discrete subset S of a topological group G with the identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We apply Michael's selection theorem to offer a direct, self-contained, purely topological proof of the result of Hofmann and Morris [K.-H. Hofmann, S.A. Morris, Weight and c, J. Pure Appl. Algebra 68 (1-2) (1990) 181-194] on the existence of suitable sets in locally compact groups. Our approach uses only elementary facts from (topological) group theory. 相似文献
12.
Mikhail Tkachenko 《Journal of Pure and Applied Algebra》2010,214(7):1103-1109
We show that every Abelian group G with r0(G)=|G|=|G|ω admits a pseudocompact Hausdorff topological group topology T such that the space (G,T) is Fréchet-Urysohn. We also show that a bounded torsion Abelian group G of exponent n admits a pseudocompact Hausdorff topological group topology making G a Fréchet-Urysohn space if for every prime divisor p of n and every integer k≥0, the Ulm-Kaplansky invariant fp,k of G satisfies (fp,k)ω=fp,k provided that fp,k is infinite and fp,k>fp,i for each i>k.Our approach is based on an appropriate dense embedding of a group G into a Σ-product of circle groups or finite cyclic groups. 相似文献
13.
Dikran Dikranjan 《Topology and its Applications》2007,154(7):1321-1340
We study CLP-compact spaces (every cover consisting of clopen sets has a finite subcover) and CLP-compact topological groups. In particular, we extend a theorem on CLP-compactness of products from [J. Steprāns, A. Šostak, Restricted compactness properties and their preservation under products, Topology Appl. 101 (3) (2000) 213-229] and we offer various criteria for CLP-compactness for spaces and topological groups, that work particularly well for precompact groups. This allows us to show that arbitrary products of CLP-compact pseudocompact groups are CLP-compact. For every natural n we construct:
- (i)
- a totally disconnected, n-dimensional, pseudocompact CLP-compact group; and
- (ii)
- a hereditarily disconnected, n-dimensional, totally minimal, CLP-compact group that can be chosen to be either separable metrizable or pseudocompact (a Hausdorff group G is totally minimal when all continuous surjective homomorphisms G→H, with a Hausdorff group H, are open).
14.
Dmitri Shakhmatov 《Topology and its Applications》2010,157(8):1518-324
Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F⊆X and every point x∈X?F, there exist f∈Cp(X,G) and g∈G?{e} such that f(x)=g and f(F)⊆{e}; (b) G?-regular provided that there exists g∈G?{e} such that, for each closed set F⊆X and every point x∈X?F, one can find f∈Cp(X,G) with f(x)=g and f(F)⊆{e}. Spaces X and Y are G-equivalent provided that the topological groups Cp(X,G) and Cp(Y,G) are topologically isomorphic.We investigate which topological properties are preserved by G-equivalence, with a special emphasis being placed on characterizing topological properties of X in terms of those of Cp(X,G). Since R-equivalence coincides with l-equivalence, this line of research “includes” major topics of the classical Cp-theory of Arhangel'ski? as a particular case (when G=R).We introduce a new class of TAP groups that contains all groups having no small subgroups (NSS groups). We prove that: (i) for a given NSS group G, a G-regular space X is pseudocompact if and only if Cp(X,G) is TAP, and (ii) for a metrizable NSS group G, a G?-regular space X is compact if and only if Cp(X,G) is a TAP group of countable tightness. In particular, a Tychonoff space X is pseudocompact (compact) if and only if Cp(X,R) is a TAP group (of countable tightness). Demonstrating the limits of the result in (i), we give an example of a precompact TAP group G and a G-regular countably compact space X such that Cp(X,G) is not TAP.We show that Tychonoff spaces X and Y are T-equivalent if and only if their free precompact Abelian groups are topologically isomorphic, where T stays for the quotient group R/Z. As a corollary, we obtain that T-equivalence implies G-equivalence for every Abelian precompact group G. We establish that T-equivalence preserves the following topological properties: compactness, pseudocompactness, σ-compactness, the property of being a Lindelöf Σ-space, the property of being a compact metrizable space, the (finite) number of connected components, connectedness, total disconnectedness. An example of R-equivalent (that is, l-equivalent) spaces that are not T-equivalent is constructed. 相似文献
15.
A. Bouziad 《Topology and its Applications》2007,154(2):287-301
A topological space X is said to have the Scorza-Dragoni property if the following property holds: For every metric space Y and every Radon measure space (T,μ), any Carathéodory function is Luzin measurable, i.e., given ε>0, there is a compact set K in T with μ(T?K)?ε such that the mapping is continuous. We present a selection of spaces without the Scorza-Dragoni property, among which there are first countable hereditarily separable and hereditarily Lindelöf compact spaces, separable Moore spaces and even countable k-spaces. In the positive direction, it is shown that every space which is an ℵ0-space and kR-space has the Scorza-Dragoni property. We also prove that every separately continuous mapping , where Y is a metric space, is Luzin measurable, provided the space X is strongly functionally generated by a countable collection of its bounded subsets. If Martin's Axiom is assumed then all metric spaces of density less than c, and all pseudocompact spaces of cardinality less than c, have the Scorza-Dragoni property with respect to every separable Radon measure μ. Finally, the class of countable spaces with the Scorza-Dragoni property is closely examined. 相似文献
16.
Roberto E. Madariaga-Garcia 《Topology and its Applications》2007,154(7):1470-1480
Tkachenko showed in 1990 the existence of a countably compact group topology on the free Abelian group of size c using CH. Koszmider, Tomita and Watson showed in 2000 the existence of a countably compact group topology on the free Abelian group of size c2 using a forcing model in which CH holds.Wallace's question from 1955, asks whether every both-sided cancellative countably compact semigroup is a topological group. A counterexample to Wallace's question has been called a Wallace semigroup. In 1996, Robbie and Svetlichny constructed a Wallace semigroup under CH. In the same year, Tomita constructed a Wallace semigroup from MAcountable.In this note, we show that the examples of Tkachenko, Robbie and Svetlichny, and Koszmider, Tomita and Watson can be obtained using a family of selective ultrafilters. As a corollary, the constructions presented here are compatible with the total failure of Martin's Axiom. 相似文献
17.
We show that it is consistent with ZFC that the free Abelian group of cardinality c admits a topological group topology that makes it countably compact with a non-trivial convergent sequence. 相似文献
18.
Denote by the family of all real valued functions on a metric space which satisfy a Lipschitz condition on the compact (bounded) subsets of X. We prove that every homomorphism on is the evaluation at some point of X if and only if X is realcompact (every closed bounded subset of X is compact).
(Received 4 November 1998; in revised form 31 May 1999) 相似文献
19.
Taras Banakh Dušan Repovš Lyubomyr Zdomskyy 《Journal of Pure and Applied Algebra》2008,212(9):2105-2114
In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan Sω in a topological group G admitting a functorial embedding [0,1]⊂G. The latter means that each autohomeomorphism of [0,1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing Sω. 相似文献
20.
Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω12 that is an HFD with the following property
- (P)
- the projection of H onto every partial product I2 for I∈ω[ω1] is onto.