Souslin Number Research Articles

The local properties in topological groups and related concepts and questions

The next general question is considered below: when certain open subspaces of topological groups are homeomorphic to some topological group? A version of this question has been posed in [4] (open problem 1.4.1). In it, the complements to the elements in any zero-dimensional topological group were considered. A result below in this direction is Theorem 3.1 saying that if G is a σ-compact topological group and F is a nonempty compact subspace of G such that the complement G∖F is homeomorphic to some topological group, then F is a Gδ-subset of G (and therefore, by a remarkable theorem of M.M. Choban [6], F is a dyadic compactum). Another conclusion in this theorem says that if, moreover, the tightness of F is countable (in particular, if F is first countable), or if F is topologically linearly orderable, then G has a countable network. Lemma 4.1 says that if X is a Lindelöf homogeneous space with countable Souslin number and X is puncturable at some point a, then a is a Gδ-point in X, the subspace Y=X∖{a} is Lindelöf, and X is submetrizable. It follows from this that every Lindelöf topological field with countable Souslin number is submetrizable (its topology contains a metrizable topology). A natural Abelian topological group with few open subspaces that are homeomorphic to topological groups is identified in Example 3.2.

Some properties of the [formula omitted]-nucleus

In this paper we study the behavior of some properties of topological spaces under the influence of the Nτφ-kernel of a space X. We prove that the Nτφ-kernel of a space X preserves the density, network and π-weight of normal spaces. Besides, it is shown that the N-compact kernel of a space X preserves the Souslin number, weight, density, and π-network weight of normal spaces.

Baire irresolvable spaces with countable Souslin number

A topological space X is almost resolvable if X is the union of a countable collection of subsets each of them with empty interior. In this paper we prove that under the Continuum Hypothesis, the existence of a measurable cardinal is equivalent to the existence of a Baire crowded ccc almost irresolvable T1 space. We also prove (1.) Every crowded ccc space with cardinality less than the first weakly inaccessible cardinal is almost resolvable. (2.) If 2ω is less than the first weakly inaccessible cardinal, then every T2 crowded ccc space is almost resolvable.

Two questions on rectifiable spaces and related conclusions

In the first part of this note, we answer two open questions on rectifiable spaces.We show that if C is a compact subset of a rectifiable space G and F is closed in G then C⋅F is closed in G. This answers a question of [F.C. Lin, R.X. Shen, On rectifiable spaces and paratopological groups, Topology Appl. 158 (2011) 597–610]. We also show that every rectifiable p-space with a countable Souslin number is Lindelöf. This gives a positive answer to a question of [A.V. Arhangelʼskii, M.M. Choban, Remainders of rectifiable spaces, Topology Appl. 157 (2010) 789–799].In the last part of this note we point out that a non-locally compact rectifiable paracompact space has the following conclusion:If a non-locally compact rectifiable paracompact space G has a compactification bG such that the remainder bG∖G of G belongs to P, then G and bG∖G are separable and metrizable, where P is a class of spaces which satisfies the following conditions:(1)if X∈P, then every compact subset of the space X is a Gδ-set of X;(2)if X∈P and X is not locally compact, then X is not locally countably compact;(3)if X∈P and X is a Lindelöf p-space, then X is metrizable.Some known conclusions on rectifiable paracompact spaces and their remainders can be gotten by this conclusion.

Reflecting topological properties in continuous images

Abstract Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ + for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ + for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight ω 1. Besides, under MA + ¬ CH, the Fréchet-Urysohn property does not reflect in continuous images of weight ω 1 even for compact spaces. An application of our techniques gives a solution of an open problem published by Ramírez-Páramo. If Jensen’s κ +-Axiom $$\left( {\diamondsuit _{\kappa ^ + } } \right)$$ holds for an infinite cardinal κ, then for an arbitrary space X with no G κ-points there exists a continuous surjective map f: X → Y such that w(Y) = κ + and Y has no G tk-points. We apply this result to solve a problem of Kalenda.

Reduced multiplication modules

An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of $\mbox{\rm Spec}(M)$ . Also a finitely generated module M is a Baer module if and only if $\mbox{\rm Spec}(M)$ is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N, $\mbox{\rm Ann}(x) \not \subseteq (N:M).$ When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM.

Some upper bounds for density of function spaces

Let C α ( X , Y ) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X which is closed under finite unions. We proved that the density of the space C α ( X , Y ) is at most iw ( X ) ⋅ d ( Y ) where iw ( X ) denotes the i-weight of the Tychonoff space X, and d ( Y ) denotes the density of the space Y when Y is an equiconnected space with equiconnecting function Ψ, and Y has a base consists of Ψ-convex subsets of Y. We also prove that the equiconnectedness of the space Y cannot be replaced with pathwise connectedness of Y. In fact, it is shown that for each infinite cardinal κ, there is a pathwise connected space Y such that π-weight of Y is κ, but Souslin number of the space C k ( [ 0 , 1 ] , Y ) is 2 κ .

On Some Properties of Essential Darboux Rings of Real Functions Defined on Topological Spaces

This paper deals with rings of real Darboux functions defined on some topological spaces. Results are given concerning the existence of essential, as well as prime Darboux rings. We prove that, under some assumptions connected with the domain $X$ of the functions, the equalities: $D(X)=S_{lf}(X), S(X)=\dim(\Re )$ hold, where $D(X)$ is a ${\cal D}$-number of $X$, $S(X)$ ($S_{lf}(X)$) denotes the Souslin (lf-Souslin) number of $X$ and $\dim(\Re )$ is a Goldie dimension of an arbitrary prime Darboux ring $\Re$.

Remainders in compactifications and generalized metrizability properties

When does a Tychonoff space X have a Hausdorff compactification with the remainder belonging to a given class of spaces? A classical theorem of Henriksen and Isbell and certain theorems, involving a new completeness type property introduced below, are applied to obtain new results on remainders of topological spaces and groups. In particular, some strong necessary conditions for a topological group to have a metrizable remainder, or a paracompact p-remainder, are established (the group itself turns out to be a paracompact p-space (Theorem 4.8)). It follows that if a non-locally compact topological group G is metrizable at infinity, then G is a Lindelöf p-space, and the Souslin number of G is countable (Corollary 4.10). This solves Problem 10.28 from [M. Hušek, J. van Mill (Eds.), Recent Progress in General Topology, vol. 2, North-Holland, 2002, pp. 1–57].

Essential ideals inC(X)

It is shown thatX is finite if and only ifC(X) has a finite Goldie dimension. More generally we observe that the Goldie dimension ofC(X) is equal to the Souslin number ofX. Essential ideals inC(X) are characterized via their corresponding z-filters and a topological criterion is given for recognizing essential ideals inC(X). It is proved that the Frechet z-filter (cofinite z-filter) is the intersection of essential z-filters. The intersection of idealsOx wherex runs through nonisolated points inX is the socle ofC(X) if and only if every open set containing all nonisolated points is cofinite. Finally it is shown that if every essential ideal inC(X) is a z-ideal thenX is a P-space.

The Souslin number in set-theoretic topology

This paper presents a short survey of some directions in the theory of cardinal functions, in which estimations containing the Souslin number play the central role. Together with the known facts, the paper contains some new results established by the author. In Section 4 three very general formulae are proved; we use these formulae in the proof of the main result of the paper (Theorem 5.17) about the strict ap-caliber.

On the relative cellularity of Lindelöf subspaces of topological groups

The notions of the relative cellularity (≡ relative Souslin number) c( X, Y) and the relative τ-cellularity cel τ ( X, Y) of a subspace X of a topological space Y are introduced. Always c( X)⩾ c( X, Y)⩽cel τ ( X, Y)⩽ cel τ ( X). It is proved that cel τ ( X, G) ⩽exp τ for any τ-Lindelöf subspace X of any Hausdorff topological group G and c( X)⩽cel τ ( X)⩽ exp τ if, in addition, X is a retract of G or even a retract of some G τ-subset of G. These results are deduced form the results concerning spaces with lattices of open mappings on them.

Souslin number of the space of subgroups of a locally compact group

Souslin number of the space of subgroups of a locally compact group

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 G 0, G 1 such that G 0 × G 1 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 X 0, X 1 such that X 0 × X 1 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.

Duality with respect to the functor Cp and cardinal invariants of the type of the Souslin number

Duality with respect to the functor Cp and cardinal invariants of the type of the Souslin number

Calibers of compact spaces

LetXXbe a compact Hausdorff space andκ\kappaits Souslin number.2^{2}We prove that ifα\alphais a cardinal such that eitherα\alphaandcf⁡(α)\operatorname {cf} (\alpha )are greater thanκ\kappaand stronglyκ\kappa-inaccessible or elseα\alphais regular and greater thanκ\kappa, thenXXhas(α,ακ⏝)(\alpha , \sqrt [\underparen {\kappa }]{\alpha })caliber. Restricting our interest to the category of compact spacesXXwithS(X)=ω+S(X) = {\omega ^ + }(i.e.XXsatisfy the countable chain condition), the above statement takes, under G.C.H., the following form. For any compact spaceXXwithS(X)=ω+S(X) = {\omega ^ + }, we have that (a) ifα\alphais a cardinal andcf⁡(α)\operatorname {cf} (\alpha )does not have the formβ+{\beta ^ + }withcf⁡(β)=ω\operatorname {cf} (\beta ) = \omega, thenα\alphais caliber for the spaceXX. (b) Ifε=β+\varepsilon = {\beta ^ + }andcf⁡(β)=ω\operatorname {cf} (\beta ) = \omegathen(α,β)(\alpha ,\,\beta )is caliber forXX. A related example shows that the result of (b) is in a sense the best possible.