Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)?τ, can act properly on Rτ?{0} such that Rτ?{0} contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ?τ. The metric cones Cone(G/H) with H⊂G a compact subgroup such that G/H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H⊂G does not exceed the weight of G.     

The D-property of some Lindelöf spaces and related conclusions

It is shown that the space Cp(τω) is a D-space for any ordinal number τ, where . This conclusion gives a positive answer to R.Z. Buzyakova's question. We also prove that another special example of Lindelöf space is a D-space. We discuss the D-property of spaces with point-countable weak bases. We prove that if a space X has a point-countable weak base, then X is a D-space. By this conclusion and one of T. Hoshina's conclusion, we have that if X is a countably compact space with a point-countable weak base, then X is a compact metrizable space. In the last part, we show that if a space X is a finite union of θ-refinable spaces, then X is a αD-space.     

Group-valued continuous functions with the topology of pointwise convergence

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.     

Fibrant extensions of free G-spaces

We show that any equivariant fibrant extension of a compact free G-space is also free. This result allows us to prove that the orbit space of any equivariant fibrant compact space E is also fibrant, provided that E has only one orbit type.     

Some generalizations of the concept of a p-space

In this article, some generalizations of the concept of a p-space are introduced and studied. The notion of a source of a space in a larger space and the concepts of partial plumage, s-embedding, p-embedding, p^?-embedding, s-space, and p^?-space are defined and studied in depth (see Theorems 2.6, 2.7, 3.2, 4.3, 4.4, 4.10 and their corollaries). An example of a hereditarily p^?-space which is not a p-space and is a perfect image of a hereditarily p-space is indicated (Example 2.9). Among the main results, we establish that if a paracompact space X is p-embedded in a pseudocompact space as a dense subspace, then X is a p-space (Corollary 4.8), and that if X has a countable network and is p^?-embedded in a pseudocompact space, then X is metrizable (Corollary 4.11). The following problem is posed: is every paracompact Gδ-subspace of a pseudocompact space ?ech-complete?     

The reduced measure algebra and a K1-space which is not K0

The reduced measure algebra is used to construct, under CH, a hereditarily Lindelöf separable K₁-space X which is not a K₀-space.     

G-compactifications of pseudocompact G-spaces

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.     

On the product of homogeneous spaces

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 X_i 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.     

Universal free G-spaces

For a compact Lie group G, we prove the existence of a universal G-space in the class of all paracompact (respectively, metrizable, and separable metrizable) free G-spaces. We show that such a universal free G-space cannot be compact.     

M3-spaces whose every point has a closure preserving outer base are M1

Let X be an M₃-space. If every point of X has a closure preserving outer base, then X is an M₁-space. This is a remarkable improvement on [2, Corollary 2.8]. If there is a point of X having no closure preserving outer base, then we have an M₃-space which is not M₁.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

Actions of totally disconnected groups and equivariant singular homology

We study equivariant singular homology in the case of actions of totally disconnected locally compact groups on topological spaces. Theorem A says that if G is a totally disconnected locally compact group and X is a G-space, then any short exact sequence of covariant coefficient systems for G induces a long exact sequence of corresponding equivariant singular homology groups of the G-space X. In particular we consider the case where G is a totally disconnected compact group, i.e., a profinite group, and G acts freely on X. Of special interest is the case where G is a p-adic group, p a prime. The conjecture that no p-adic group, p a prime, can act effectively on a connected topological manifold, is namely known to be equivalent to the famous Hilbert-Smith conjecture. The Hilbert-Smith conjecture is the statement that, if a locally compact group G acts effectively on a connected topological manifold M, then G is a Lie group.     

Monolithic spaces and D-spaces revisited

We introduce the classes of monotonically monolithic and strongly monotonically monolithic spaces. They turn out to be reasonably large and with some nice categorical properties. We prove, in particular, that any strongly monotonically monolithic countably compact space is metrizable and any monotonically monolithic space is a hereditary D-space. We show that some classes of monolithic spaces which were earlier proved to be contained in the class of D-spaces are monotonically monolithic. In particular, Cp(X) is monotonically monolithic for any Lindelöf Σ-space X. This gives a broader view of the results of Buzyakova and Gruenhage on hereditary D-property in function spaces.     

On isometrically universal spaces, mappings, and actions of groups

In this paper we first consider some well-known classes of separable metric spaces which are isometrically ω-saturated (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559]) and, therefore, contain isometrically universal spaces. We put some problems concerning such spaces most of which are related with the properties of the isometrically universal Urysohn space. Furthermore, using the defined notions of isometrically universal mappings and G-spaces (which are analogies of the notion of isometrically universal spaces) we introduce the notions of an isometrically ω-saturated class of mappings and an isometrically ω-saturated class of G-spaces (in which there are “many” isometrically universal elements). We prove that all results of Sections 6.1 and 7.1 of [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559] can be reformulated for isometrically ω-saturated classes of spaces and G-spaces, respectively. In particular, we prove that if D and R are isometrically ω-saturated classes of spaces, then the class of all mappings with the domain in D and range in R is an isometrically ω-saturated class of mappings and, therefore, in this class there are isometrically universal elements. As a corollary of this result we have that since the class of all mappings is isometrically ω-saturated, in this class there are isometrically universal mappings. Similarly, if G is an arbitrary separable metric group and P is an isometrically ω-saturated class of spaces, then the class of all G-spaces (X,F), where X is an element of P, is an isometrically ω-saturated class of G-spaces and, therefore, in this class there are isometrically universal elements. In particular, for any separable metric group G, in the class of all G-spaces there are isometrically universal G-spaces. We also pose some problems concerning isometrically universal mappings and G-spaces some of which concern the Urysohn space.     

On finite unions of certain D-spaces

In this note, we show that if X is the union of a finite collection of strong Σ-spaces, then X is a D-space. As a corollary, we get a conclusion that if X is the union of a finite collection of Moore spaces, then X is a D-space. This gives a positive answer to one of Arhangel'skii's problems [A.V. Arhangel'skii, D-spaces and finite unions, Proc. AMS 132 (7) (2004) 2163-2170]. In the last part of the note, we show that if X is the union of a finite collection of DC-like spaces, then X is a D-space, where DC is the class of all discrete unions of compact spaces. As a corollary, we show that if X is the union of a finite collection of regular subparacompact C-scattered spaces, then X is a D-space.     

Equivariant strong shape

In this paper we propose a construction of the equivariant strong shape for compact metrizable G-spaces using an equivariant version of so-called cotelescopes and the concept of a fibrant G-space.     

Spaces in which every dense subset is a Gδ

A topological space X is called a DG_δ-space if every subset of X is a G_δ-set in its closure. In this paper we study DG_δ-spaces that contains subspaces in which every dense subset is open and spaces in which every subset is a G_δ . We give some new results in these classes of topological spaces.