Tychonoff Product Research Articles

Embedding theorems for function spaces

In this paper, we have proved results similar to Tychonoff's Theorem on embedding a space of functions with the topology of pointwise convergence into the Tychonoff product of topological spaces, but applied to the function space C(X,Y) of all continuous functions from a topological space X into a uniform space Y with the topology of uniform convergence on a family of subsets of X and with the (weak) set-open topology. We also investigated the following question: how the topological embedding of the space C(X,Y) is related to algebraic structures (such as topological groups, topological rings and topological vector spaces) on C(X,Y).

On butterfly-points in $\beta X$, Tychonoff products and weak Lindelöf numbers

Let $X$ be the Tychonoff product $\prod _{\alpha <\tau}X_{\alpha}$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha}$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \{p\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X=\omega^{\tau}$ or $X=R^{\tau}$ and $\tau$ is infinite and not countably cofinal.

Some dense sets of the Tychonoff cube Ic

By Hewitt–Marczewski–Pondiczery theorem (see [3]) the Tychonoff product of 2ω many separable spaces ▪ is separable.The problem of the existence of countable dense sets in a product of separable spaces ▪ with additional properties is very important.We prove that in the Tychonoff cube Ic there are dense sets such that all their countable subsets have properties, which entail, in particular, that these dense sets contain no convergent sequences.

On non-normality points, Tychonoff products and Suslin number

Let a space $X$ be Tychonoff product $\prod_{\alpha <\tau}X_{\alpha}$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha}$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech--Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau}$ or $\omega ^{\tau}$ and a cardinal $\tau$ is infinite and not countably cofinal.

PRODUCTS OF ULTRAFILTERS AND MAXIMAL LINKED SYSTEMS ON WIDELY UNDERSTOOD MEASURABLE SPACES

Constructions related to products of maximal linked systems (MLSs) and MLSs on the product of widely understood measurable spaces are considered (these measurable spaces are defined as sets equipped with \(\pi\)-systems of their subsets; a \(\pi\)-system is a family closed with respect to finite intersections). We compare families of MLSs on initial spaces and MLSs on the product. Separately, we consider the case of ultrafilters. Equipping set-products with topologies, we use the box-topology and the Tychonoff product of Stone-type topologies. The properties of compaction and homeomorphism hold, respectively.

Uniformities on strongly topological gyrogroups

In this paper, we introduced three uniformities VGl, VGr and VG which are induced in a natural way on a strongly topological gyrogroup (G,⊕,τ). It is mainly proved that (1) each of the three uniformities is compatible with G; (2) if H is a subgyrogroup of G, then VG,Hl=VHl, VG,Hr=VHr and VG,H=VH; (3) if {(Gi,⊕i,τi):i∈I} is a family of strongly topological gyrogroups, then VGl=∏i∈IVGil,VGr=∏i∈IVGir and VG=∏i∈IVGi, where G=∏i∈IGi endowed with the Tychonoff product topology. At the end section, we obtain that every compact strongly topological gyrogroup has property U and every Lindelöf strongly topological gyrogroup has property ω-U.

Topological groups with invariant linear spans

Given a topological group G that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that G has invariant linear span if all linear spans of G under arbitrary embeddings into topological vector spaces are isomorphic as topological vector spaces. For an arbitrary set A let \({{\mathbb {Z}}}^{(A)}\) be the direct sum of |A|-many copies of the discrete group of integers endowed with the Tychonoff product topology. We show that the topological group \({{\mathbb {Z}}}^{(A)}\) has invariant linear span. This answers a question from a paper of Dikranjan et al. (J Math Anal Appl 437:1257–1282, 2016) in positive. We prove that given a non-discrete sequential space X, the free abelian topological group A(X) over X is an example of a topological group that embeds into a topological vector space but does not have invariant linear span.

On the Topological Structure and Properties of Multidimensional (C, R) Space

Generally, the linear topological spaces successfully generate Tychonoff product topology in lower dimensions. This paper proposes the construction and analysis of a multidimensional topological space based on the Cartesian product of complex and real spaces in continua. The geometry of the resulting space includes a real plane with planar rotational symmetry. The basis of topological space contains cylindrical open sets. The projection of a cylindrically symmetric continuous function in the topological space onto a complex planar subspace maintains surjectivity. The proposed construction shows that there are two projective topological subspaces admitting non-uniform scaling, where the complex subspace scales at a higher order than the real subspace generating a quasinormed space. Furthermore, the space can be equipped with commutative and finite translations on complex and real subspaces. The complex subspace containing the origin of real subspace supports associativity under finite translation and multiplication operations in a combination. The analysis of the formation of a multidimensional topological group in the space requires first-order translation in complex subspace, where the identity element is located on real plane in the space. Moreover, the complex translation of identity element is restricted within the corresponding real plane. The topological projections support additive group structures in real one-dimensional as well as two-dimensional complex subspaces. Furthermore, a multiplicative group is formed in the real projective space. The topological properties, such as the compactness and homeomorphism of subspaces under various combinations of projections and translations, are analyzed. It is considered that the complex subspace is holomorphic in nature.

The weight of lexicographic products

We will calculate the weight of lexicographic products of GO-spaces, using this we will see:•the assertion that the weight of the lexicographic product 2ω1 is ℵ1 is equivalent to the Continuum Hypothesis (CH), that is, 2ℵ0=ℵ1,•the assertion that the weight of both lexicographic products 2ω1 and 2ω1+1 coincide is equivalent to the assertion 2ℵ0=2ℵ1,•the assertion that the lexicographic product 2γ is homeomorphic to the usual Tychonoff product 2γ is equivalent to γ≤ω.

Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids

We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

Sequences and dense sets

By the classical Hewitt–Marczewski–Pondiczery theorem (see [2], [3]) the Tychonoff product of 2ω many separable spaces is separable [2,3].We consider the problem of the existence in the Tychonoff product of 2ω many separable spaces a dense countable subset Q, which contains no nontrivial convergent sequences.We prove (Theorem 4.1) that such dense set exists in the product ∏α∈2ωZα of separable spaces {Zα:α∈2ω} if in every space Zα (α∈2ω) there are two closed disjoint not empty sets, we call such spaces decomposable.The class of decomposable spaces includes not single point T1-spaces, but also some T0-spaces and some spaces, which are not even T0-spaces.

On the density and products of spaces

We consider problems, connected with the Hewitt–Marczewski–Pondiczery theorem on dense subsets of the Tychonoff product of spaces. We consider the problem of the existence of dense subsets of ∏α∈AZα which have additional properties. There are some important results on the products of 2ω many separable spaces. We consider the product of spaces which have the uncountable density.By using some (2τ,τ)-independent matrix, we prove (Theorem 3.2) that if {Zα:α∈2τ} is a family of spaces for an uncountable cardinal τ, where d(Zα)=τ for all α∈2τ, then in ∏α∈2τZα there is a dense set Q, |Q|=τ, such that for every PQ⊆Q, |PQ|=τ, α-projection πα(PQ) is dense in Zα for some α∈2τ.

A Generalization of Tychonoff's Product Theorem

A set-theoretical generalization of Tychonoff's theorem on compactness of the product of compact topological spaces is proved.

Supercompact minus compact is super

According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed k-network. This characterization suggests to call a topological space super if it has a binary closed k-network N. The binarity of N means that every linked subfamily of N is centered. Therefore, a Hausdorff space is supercompact if and only if it is super and compact. In this paper we prove that the class of super spaces contains all GO-spaces, all supercompact spaces, all metrizable spaces, and all collectionwise normal ℵ-spaces. Moreover, the class of super spaces is closed under taking Tychonoff products and discretely dense sets in Tychonoff products. The superness of metrizable spaces implies that each compact metrizable space is supercompact, which was first proved by Strok and Szymański (1975) and then reproved by Mills (1979), van Douwen (1981), and Dȩbski (1984).

On dense subsets of Tychonoff products of T1-spaces

We consider dense subsets of Zc=∏α∈2ωZα, where Z=Zα is a separable not single point T1-space.We construct in Zc, |Z|≥ω, (Theorem 4.1) a countable dense set Q⊆Z such that every countable subset of Q contains a countable subset, which can be project on “many” subsets of Z.From this theorem follow some facts.Theorem 4.2 states that in the product Zc of a not single point separable T1-space Z there is a countable dense set which contains no non-trivial convergent in Zc sequences.The existence of such set in the product Ic, where I=[0,1] was proved [13] by W. H. Priestley.In [9] we proved that such set exists in a product of 2ω separable not single point T2-spaces.Theorem 4.4 states that if Z is a separable not countably compact T1-space, then there is a countable dense subset Q⊆Zc, satisfying the following condition: if E⊆Q is a countable set and E converges to a set F⊆Zc, then |E∖F|<ω.

ON THE SET-THEORETIC STRENGTH OF ELLIS’ THEOREM AND THE EXISTENCE OF FREE IDEMPOTENT ULTRAFILTERS ON ω

AbstractEllis’ Theorem (i.e., “every compact Hausdorff right topological semigroup has an idempotent element”) is known to be proved only under the assumption of the full Axiom of Choice (AC); AC is used in the proof in the disguise of Zorn’s Lemma.In this article, we prove that in ZF, Ellis’ Theorem follows from the Boolean Prime Ideal Theorem (BPI), and hence is strictly weaker than AC in ZF. In fact, we establish that BPI implies the formally stronger (than Ellis’ Theorem) statement “for every family ${\cal A} = \{ ({S_i},{ \cdot _i},{{\cal T}_i}):i \in I\}$ of nontrivial compact Hausdorff right topological semigroups, there exists a function f with domain I such that $f\left( i \right)$ is an idempotent of ${S_i}$, for all $i \in I$”, which in turn implies ACfin (i.e., AC for sets of nonempty finite sets).Furthermore, we prove that in ZFA, the Axiom of Multiple Choice (MC) implies Ellis’ Theorem for abelian semigroups (i.e., “every compact Hausdorff right topological abelian semigroup has an idempotent element”) and that the strictly weaker than MC (in ZFA) principle LW (i.e., “every linearly ordered set can be well-ordered”) implies Ellis’ Theorem for linearly orderable semigroups (i.e., “every compact Hausdorff right topological linearly orderable semigroup has an idempotent element”); thus the latter formally weaker versions of Ellis’ Theorem are strictly weaker than BPI in ZFA. Yet, it is shown that no choice is required in order to prove Ellis’ Theorem for well-orderable semigroups.We also show that each one of the (strictly weaker than AC) statements “the Tychonoff product $2^{\Cal R} $ is compact and Loeb” and $BPI_{\Cal R}$ (BPI for filters on ${\Cal R}$) implies “there exists a free idempotent ultrafilter on ω” (which in turn is not provable in ZF). Moreover, we prove that the latter statement does not imply $BP{I_\omega }$ (BPI for filters on ω) in ZF, hence it does not imply any of $AC_{\Cal R} $ (AC for sets of nonempty sets of reals) and $BPI_{\Cal R} $ in ZF, either.In addition, we prove that the statements “there exists a free ultrafilter on ω”, “there exists a free ultrafilter on ω which is not idempotent”, and “for every IP set $A \subseteq \omega$, there exists a free ultrafilter ${\cal F}$ on ω such that $A \in {\cal F}$” are pairwise equivalent in ZF.

On dense subsets, convergent sequences and projections of Tychonoff products

It is well know that the Tychonoff product of 2ω many separable spaces is separable [2,3].We consider for the Tychonoff product of 2ω many separable spaces the problem of the existence of a dense countable subset, which contains no nontrivial convergent in the product sequences.The first result was proved by W.H. Priestley. He proved [14] that such dense set exists in the Tychonoff product ∏α∈2ωIα of closed unit intervals.We prove (Theorem 3.2) that such dense set exists in the Tychonoff product ∏α∈2ωZα of 2ω many Hausdorff separable not single point spaces.We prove that in ∏α∈2ωZα there is a countable dense set Q⊆∏α∈2ωZα such that for every countable subset S⊆Q a set πA(S) is dense in a face ∏α∈AZα for some A, |A|=ω.We prove (Theorem 3.4) that in ∏α∈2ωIα there is a countable set, that is dense but sequentially closed in ∏α∈2ωIα with the Tychonoff topology and is closed and discrete in ∏α∈2ωIα with the box topology (Theorem 3.4).

Pseudocompactness, Products, and Topological Brandt λ0 -Extensions of Semitopological Monoids

In the present paper, we study the preservation of pseudocompactness (resp., countable compactness, sequential compactness, ω -boundedness, totally countable compactness, countable pracompactness, sequential pseudocompactness) by Tychonoff products of pseudocompact (and countably compact) topological Brandt $$ {\lambda}_i^0 $$ -extensions of semitopological monoids with zero. In particular, we show that if $$ \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} $$ is a family of Hausdorff pseudocompact topological Brandt $$ {\uplambda}_i^0 $$ -extensions of pseudocompact semitopological monoids with zero such that the Tychonoff product $$ \prod \left\{{S}_i:i\in \mathrm{\mathcal{I}}\right\} $$ is a pseudocompact space, then the direct product $$ \prod \left\{\left({B}_{\uplambda_i}^0\left({S}_i\right),\kern0.5em {\uptau}_{B\left({S}_i\right)}^0\right):i\in \mathrm{\mathcal{I}}\right\} $$ endowed with the Tychonoff topology is a Hausdorff pseudocompact semitopological semigroup.

About dense subsets of Tychonoff products of discrete spaces

The Hewitt–Marczewski–Pondiczery theorem states that if X=▪ is the Tychonoff product, where d(Xα)≤τ≥ω for all α∈A and |A|≤2τ, then d(X)≤τ.We prove that in the product ▪ of discrete spaces Dα=ω there is a dense countable subset Q=∪{Qk:k∈ω}, the union of disjoint finite sets Qk, satisfying the following conditions:–if F⊆Q is such that |F∩Qk|≤1 for all k∈ω, then F is a discrete closed in ▪ set;–if C⊆ω is infinite and for F⊆Q there is m0∈ω such that |Qk∩F|≤m0 for all k∈ω, then ∪{Qk:k∈C}∖F is dense in ▪;–Q contains no convergent in ▪ non-trivial sequences;

Some properties of topological groups related to compactness

In the present article four problems from the A.V. Arhangel'skii and M.G. Tkachenko's book [8] are examined. Theorem 2.5 affirms that for any uncountable cardinal τ there exists a zero-dimensional hereditarily paracompact non-metrizable Abelian topological group G of the weight τ1=sup{2m:m<τ} which has a linearly ordered compactification bG of countable dyadicity index. In this connection, in Section 2 we present some properties of continuous images of Tychonoff product of compact spaces of the fixed weight τ. These spaces are called τ-dyadic. By virtue of Corollary 3.3, if G is a non-metrizable topological group of pointwise countable type, then the space Ge=G∖{e} is not homeomorphic to a topological group. Section 3 contains also other results of that kind. In Section 4 some sufficient conditions are presented, under which the compact Gδ-subset of the quotient space G/H is a Dugundji space.