Normal Hausdorff Research Articles

Properties of Topologies for the Continuous Representability of All Weakly Continuous Preorders

We investigate properties of strongly useful topologies, i.e., topologies with respect to which every weakly continuous preorder admits a continuous order-preserving function. In particular, we prove that a topology is strongly useful provided that the topology generated by every family of separable systems is countable. Focusing on normal Hausdorff topologies, whose consideration is fully justified and not restrictive at all, we show that strongly useful topologies are hereditarily separable on closed sets, and we identify a simple condition under which the Lindelöf property holds.

Prime spectrums of EQ-algebras

Abstract The main purpose of this paper is to study prime spectrums of EQ-algebras and to solve two open problems about $\wedge $-prime spectrums of involutive and prelinear EQ-algebras, which were proposed by N. Akhlaghinia, R.A. Borzooei and M. A. Kologani. In order to do so, we first give some characterizations of preideals, prime preideals and maximal preideals on (good) EQ-algebras, respectively. Then we introduce the notion of quasi De Morgan EQ-algebras (MEQ-algebras for short) and obtain that $\wedge $-prime preideals coincide with prime preideals for MEQ-algebras, and each involutive EQ-algebra is an MEQ-algebra. Following, we show that the prime spectrum space of a good EQ-algebra is a compact topological space and obtain that for any involutive EQ-algebra the prime spectrum space is connected if and only if its Boolean center is indeed 2-element. Also, we prove that the maximal spectrum space of a good and prelinear EQ-algebra (or an involutive and prelinear EQ-algebra) is a normal Hausdorff space. These results totally answer the above two open problems. Finally, we give some characterizations of the spectrum space of an MEQ-algebra by its reticulation.

The finitely generated Hausdorff spectra of a family of pro-p groups

Recently the first example of a family of pro-p groups, for p a prime, with full normal Hausdorff spectrum was constructed. In this paper we further investigate this family by computing their finitely generated Hausdorff spectrum with respect to each of the five standard filtration series: the p-power series, the iterated p-power series, the lower p-series, the Frattini series and the dimension subgroup series. Here the finitely generated Hausdorff spectra of these groups consist of infinitely many p-adic rational numbers, and their computation requires a rather technical approach. This result also gives further evidence to the non-existence of a finitely generated pro-p group with uncountable finitely generated Hausdorff spectrum.

A pro-2 group with full normal Hausdorff spectra

Abstract We construct a 2-generated pro-2 group with full normal Hausdorff spectrum [ 0 , 1 ] [0,1] , with respect to each of the four standard filtration series: the 2-power series, the lower 2-series, the Frattini series, and the dimension subgroup series. This answers a question of Klopsch and the second author, for the even prime case; the odd prime case was settled by the first author and Klopsch. Also, our construction gives the first example of a finitely generated pro-2 group with full Hausdorff spectrum with respect to the lower 2-series.

A pro‐pgroup with full normal Hausdorff spectra

AbstractFor each odd prime p, we produce a 2‐generated pro‐pgroupGwhose normal Hausdorff spectrawith respect to five standard filtration series , namely the lowerp‐series, the dimension subgroup series, thep‐power series, the iteratedp‐power series and the Frattini series, are all equal to the full unit interval [0,1]. Here denotes the Hausdorff dimension function associated to the natural translation‐invariant metric induced by the filtration series .

A pro-p group with infinite normal Hausdorff spectra

Using wreath products, we construct a finitely generated pro-p group G with infinite normal Hausdorff spectrum with respect to the p-power series. More precisely, we show that this normal Hausdorff spectrum contains an infinite interval; this settles a question of Shalev. Furthermore, we prove that the normal Hausdorff spectra of G with respect to other filtration series have a similar shape. In particular, our analysis applies to standard filtration series such as the Frattini series, the lower p-series and the modular dimension subgroup series. Lastly, we pin down the ordinary Hausdorff spectra of G with respect to the standard filtration series. The spectrum of G for the lower p-series displays surprising new features.

Pro‐p groups of positive rank gradient and Hausdorff dimension

Let $G$ be a finitely generated pro-$p$ group of positive rank gradient. Motivated by the study of Hausdorff dimension, we show that finitely generated closed subgroups $H$ of infinite index in $G$ never contain any infinite subgroups $K$ that are subnormal in~$G$ via finitely generated successive quotients. This pro-$p$ version of a well-known theorem of Greenberg generalises similar assertions that were known to hold for non-abelian free pro-$p$ groups, non-soluble Demushkin pro-$p$ groups and other related pro-$p$ groups. The result we prove is reminiscent of Gaboriau's theorem for countable groups with positive first $\ell^2$-Betti number, but not quite a direct analogue. The approach via the notion of Hausdorff dimension in pro-$p$ groups also leads to our main results. We show that every finitely generated pro-$p$ group $G$ of positive rank gradient has full Hausdorff spectrum $\text{hspec}^\mathcal{F}(G) = [0,1]$ with respect to the Frattini series~$\mathcal{F}$. Using different, Lie-theoretic techniques we also prove that finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups $G$ have full Hausdorff spectrum $\text{hspec}^\mathcal{Z}(G) = [0,1]$ with respect to the Zassenhaus series~$\mathcal{Z}$. This resolves a long-standing problem in the subject of Hausdorff dimensions in pro-$p$ groups. The results about full Hausdorff spectra hold more generally for finite direct products of finitely generated pro-$p$ groups of positive rank gradient and for mixed finite direct products of finitely generated non-abelian free pro-$p$ groups and non-soluble Demushkin groups, respectively. Analogously, the aforementioned results with respect to the Frattini series generalise further to Hausdorff dimension functions with respect to arbitrary iterated verbal filtrations. Finally, we determine the normal Hausdorff spectra of such direct products.

On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Abstract Hausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in L 2 ⁢ ( ℝ n ) {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space L 2 ⁢ ( ℝ n ; ℂ 2 n ) {L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})} . Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.

On the stricture of normal Hausdorff operators on Lebesgue spaces

Мы рассматриваем обобщенные хаусдорфовы операторы и вводим для них понятие символа. Используя это понятие, мы при некоторых естественных условиях описываем структуру и исследуем важные свойства (такие, как обратимость, спектр, норма и компактность) нормальных обобщенных хаусдорфовых операторов в пространствах Лебега над $\mathbb{R}^n$. В качестве примеров рассмотрены операторы Чезаро.

Approximation by Polyhedral $G$ Chains in Banach Spaces

In a Banach space with the metric approximation property, each compactly supported rectifiable G chain whose boundary is rectifiable as well, is approximatable in the flat norm by a polyhedral G chain of nearly the same normal Hausdorff mass.

Upper semifinite hyperspaces as unifying tools in normal Hausdorff topology

In this paper we use the upper semifinite topology in hyperspaces to get results in normal Hausdorff topology. The advantage of this point of view is that the upper semifinite topology, although highly non-Hausdorff, is very easy to handle. By this way we treat different topics and relate topological properties on spaces with some topological properties in hyperspaces. This hyperspace is, of course, determined by the base space. We prove here some reciprocals which are not true for the usual Vietoris topology. We also point out that this framework is a very adequate one to construct the Čech–Stone compactification of a normal space. We also describe compactness in terms of the second countability axiom and of the fixed point property. As a summary we relate non-Hausdorff topology with some facts in the core of normal Hausdorff topology. In some sense, we reinforce the unity of the subject.

Compactifications of quasi-uniform hyperspaces

Several results on compactification of quasi-uniform hyperspaces are obtained. For instance, we prove that if C 0(X) denotes the family of all nonempty closed subsets of a quasi-uniform space (X, U) and U H the Bourbaki quasi-uniformity of U, then ( C 0(X), U H) is ∗-compactifiable if and only if (X, U) is closed symmetric and ∗-compactifiable and U −1 is hereditarily precompact. We deduce that for any normal Hausdorff space X, 2 βX is equivalent to the ∗-compactification of ( C 0(X), PN H) , where PN denotes the Pervin quasi-uniformity of X.

Γ-Representation of variational s-limits of constrained minimization problems in normal hausdorff spaces

Γ-Representation of variational s-limits of constrained minimization problems in normal hausdorff spaces

Subspaces of connected spaces

A connectification of a topological space X is a connected Hausdorff space that contains X as a dense subspace. Watson and Wilson have noted that a Hausdorff space with a connectification has no nonempty proper clopen H-closed subspaces. Here it is proven that a Hausdorff space in which every nonempty proper clopen set is not feebly compact and the cardinality of the set of clopen sets is at most 2 c is connectifiable. This result is used to show that every metric space with no nonempty proper clopen H-closed subspace is connectifiable, answering a question asked by Watson and Wilson. Also, there is a nonconnectifiable, Hausdorff space of cardinality c with no proper H-closed subspace. Using the set-theoretic hypothesis p = c, an example of a nonconnectifiable, normal Hausdorff space of cardinality c is constructed which has no nonempty compact open subset. This space is locally compact at all but one point, and if the continuum hypothesis is assumed it is first countable. This space provides a solution to questions asked by Watson and Wilson as well as Mack. The paper concludes by examining when extremally disconnected Tychonoff spaces have Tychonoff connectifications.

An Elementary Extension of Tietze's Theorem

Here C(X) and C(A) are the sets of continuous real-valued functions on X and A, respectively. F is an extension of f means F(x) =f(x) if x E A. While Theorem T is included in almost any text on point-set topology, none of the many books we surveyed mentions anything more general than Theorem T, either as a corollary or as an exercise. Of course, some texts observe that by translation one can extend a function f satisfying M1 <f(x) < M2, x E A, to a ftinction F satisfying M1 < F(x) < M2, x E X when M1 and M2 are any two constants, not just M2 = c = M as given in Theorem T. It should be observed that the original Tietze Theorem was stated for metric spaces and later generalized by Urysohn to normal Hausdorff spaces. Also, some extensions of Theorem T different from what is presented here appear in the literature (cf. [2]). This note considers the following question: Let A be a closed subset of X and assume g, h E C(X) such that g(x) < h(x) for every x E X. If f is a function in C(A) lying between g and h, i.e. g(x) <f(x) < h(x) for every x in A, can f be extended to a function F in C(X) lying between the same two functions? The following easy exercises lead directly from Theorem T to an affirmative answer to the question. However, we show later that there is a more general version that can be verified immediately from Theorem T.

A Characterization of the Topological Dimension

AbstractThis paper gives a new characterization of the dimension of a normal Hausdorff space, which joins together the Eilenberg-Otto characterization and the characterization by finite coverings. The link is furnished by the notion of a system of faces of a certain type (N1,..., NK), where N1,..., NK, K are natural numbers. It is shown that a space X contains a system of faces of type (N1,..., NK) if and only if dim(X) ≥ N1 + … + NK. The two limit cases of the theorem, namely Nk = 1 for 1 ≤ k ≤ K on the one hand, and K = 1 on the other hand, give the two known results mentioned above.

A normal collectionwise Hausdorff, not collectionwise normal space

A normal collectionwise Hausdorff, not collectionwise normal space

A set-theoretic proposition implying the metrizability of normal Moore spaces

A set-theoretic proposition is shown to be equivalent to a topological statement implying that all first countable normal Hausdorff spaces are collectionwise normal, and hence that all normal Moore spaces are metrizable.

A separable normal nonparacompact space

A topological space X is said to be paracompact [1] if for every open covering G of X there is a locally finite open covering G' of X which is a refinement of G. (G' is locally finite if every point of X has a neighborhood which intersects only a finite number of members of G'.) It is known that every paracompact Hausdorff space is normal [1 ] and that every metrizable space is paracompact [2 ]. Since every normal Hausdorff space with a countable base is metrizable, therefore, every normal Hausdorff space with a countable base is paracompact. The purpose of this paper is to show that the existence of a countable base cannot be replaced by separability in this last statement.

Countable paracompactness in linearly ordered spaces

A Hausdorff space X is said to be paracompact [2] provided that if W is a collection of open sets covering X, there exists a collection W' of open sets covering X such that (1) each element of W' is a subset of an element of W and (2) each point of X belongs to an open set which intersects only a finite number of elements of W'; X is said to be countably paracompact [3] provided that if W is a countable collection of open sets covering X, there exists a collection W' satisfying the above conditions. It is known that not every normal Hausdorff space is paracompact [2], but the question whether every such space is countably paracompact is as yet unsolved (cf. [3]). Since every linearly ordered space' is a normal Hausdorff space (cf. [1, p. 39]) but not necessarily paracompact [2], it seems natural to inquire whether every linearly ordered space must be countably paracompact. The purpose of the present note is to show that this is the case. DEFINITIONS. 1. A collection G of subsets of a space X is said to be locallyfinite provided every point of X belongs to an open set X which intersects at most a finite number of the elements of G. 2. If G and H are collections of sets, then H is said to be a refinement of G provided each element of H is a subset of some element of G. 3. A collection G of sets is said to be coherent provided G is not the sum of two collections G, and G2 such that no element of G1 intersects an element of G2. 4. If p belongs to some element of the collection G of sets, then the star of p with respect to G is the sum of the elements of G which contain p. Suppose X is a linearly ordered space and W is a countable collection of open sets covering X. For each point p of X, let Mp denote the set of all points x of X such that there is a finite, coherent collection HX of open intervals of X such that H. is a refinement of W and covers the set whose elements are p and x. For each point p of X, let Kp denote the collection of all open intervals k of X such that k is a subset of Mp and of some element of W. It is easily seen that for each p in X, Mp is both open and closed and if q M, then M = M,.