On universal spaces and absorbing sets related to a transfinite extension of covering dimension

R. Pol has shown that for every countable ordinal number α there exists a universal space for separable metrizable spaces X with trindX?α. W. Olszewski has shown that for every countable limit ordinal number λ there is no universal space for separable metrizable space with trIndX?λ. T. Radul and M. Zarichnyi have proved that for every countable limit ordinal number there is no universal space for separable metrizable spaces with dimWX?α where dimW is a transfinite extension of covering dimension introduced by P. Borst. We prove the same result for another transfinite extension dimC of the covering dimension.As an application, we show that there is no absorbing sets (in the sense of Bestvina and Mogilski) for the classes of spaces X with dimCX?α belonging to some absolute Borel class.     

There is no upper bound of small transfinite compactness degree in metrizable spaces

In [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993], Aarts and Nishiura investigated several types of dimensions modulo a class P of spaces. These dimension functions have natural transfinite extensions. The small transfinite compactness degree trcmp is such transfinite dimension function extending the small compactness degree cmp. We shall prove that there is no upper bound for trcmp in the class of metrizable spaces, i.e. for each ordinal number α there exists a metrizable space Xα such that trcmpXα=α. We also give a characterization of the dimension dim of a separable (compact) metrizable space in terms of the function cmp of the product of this space with a nowhere locally compact zero-dimensional factor.     

On negative limit sets for one-dimensional dynamics

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every α-limit set is an ω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all α-limit sets of interval maps is not closed in the Hausdorff metric (and thus some ω-limit sets are never obtained as α-limit sets). Moreover, we prove that the set of all recurrent points is closed if and only if the space of all α-limit sets is closed.     

A local duality principle for the Baire classes of functions

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. We show that, for every ordinal 1?α?ω₁, the spaces Bα[0,1] of bounded Baire functions of class α are local dual spaces of the space M[0,1] of all Borel measures. As a consequence, we derive that each annihilator Bα^⊥[0,1] is the kernel of a norm-one projection.     

A Metric Space with Transfinite Asymptotic Dimension 2ω + 1*

The authors construct a metric space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both 2ω + 1, where ω is the smallest infinite ordinal number. Therefore, an example of a metric space with asymptotic property C is obtained.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

Ordered spaces, metric preimages, and function algebras

We consider the Complex Stone-Weierstrass Property (CSWP), which is the complex version of the Stone-Weierstrass Theorem. If X is a compact subspace of a product of three linearly ordered spaces, then X has the CSWP if and only if X has no subspace homeomorphic to the Cantor set. In addition, every finite power of the double arrow space has the CSWP. These results are proved using some results about those compact Hausdorff spaces which have scattered-to-one maps onto compact metric spaces.     

On the base dimension I and the property of universality

In Iliadis (2005) [13] for an ordinal α the notion of the so-called (bn-Ind?α)-dimensional normal base C for the closed subsets of a space X was introduced. This notion is defined similarly to the classical large inductive dimension Ind. In this case we shall write here I(X,C)?α and say that the base dimension I of the space X by the normal base C is less than or equal to α. The classical large inductive dimension Ind of a normal space X, the large inductive dimension Ind₀ of a Tychonoff space X defined independently by Charalambous and Filippov, as well as, the relative inductive dimension defined by Chigogidze for a subspace X of a Tychonoff space Y may be considered as the base dimension I of X by normal bases Z(X) (all closed subsets of X), Z(X) (all functionally closed subsets of X), and , respectively.In the present paper, we shall consider normal bases of spaces consisting of functionally closed subsets. In particular, we introduce new dimension invariant : for a space X, is the minimal element α of the class O∪{−1,∞}, where O is the class of all ordinals, for which there exists a normal base C on X consisting of functionally closed subsets such that I(X,C)?α. We prove that in the class of all completely regular spaces X of weight less than or equal to a given infinite cardinal τ such that there exist universal spaces. However, the following questions are open.(1) Are there universal elements in the class of all normal (respectively, of all compact) spaces X of weight ?τ with ?(2) Are there universal elements in the class of all Tychonoff (respectively, of all normal) spaces X of weight ?τ with Ind₀(X)?n∈ω? (Note that for a compact space X.)     

Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map , T is a topological space, such that for each the set g ^-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality for a large class of metric spaces X including all locally compact Hadamard spaces, where rank _h X is the maximal topological dimension of among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank _h conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H ⁿ with . Submitted: February 2001, Revised: October 2001.     

On the role of finite,hereditarily normal spaces and maps in the genesis of compact Hausdorff spaces

We consider how properties of the bonding maps of the inverse spectrum determine properties of the inverse limit. Specifically, we study the limits of inverse spectra of finite T₀-spaces with bonding maps which are either chaining or normalizing. We will show that if the bonding maps are normalizing, then the inverse limit is a normal T₀-space, and therefore, its Hausdorff reflection is its subset of specialization minimal elements. If the maps are chaining, then the inverse limit is a completely normal spectral space; such spaces have been studied since they include the real spectra of commutative rings [C.N. Delzell, J.J. Madden, J. Algebra 169 (1994) 71], and the prime spectrum of a ring of functions, Spec(C(X)). The existence and importance of this class of non-Hausdorff, normal topological spaces was extremely surprising to us. Further, each of these results is reversible; if the inverse limit is normal, then each space in the spectrum is preceded by one whose bonding map to it is normalizing. By way of contrast, the inverse limit of finite T₀-spaces with separating bonding maps need not be a normal topological space (Example 3.8(a)) and furthermore, if the spaces of the inverse spectrum are normal, then the Hausdorff reflection of the limit must be zero-dimensional (Theorem 3.15).     

Boundedness, one-point extensions and B-extensions

The general concept of boundedness in a topological space generalizes both metric boundedness and relative compactness. A one-point extension o(ℱ _X) of the space X is naturally associated to each boundedness ℱ _X and every Hausdorff one-point extensions of a space X can be obtained in this way. Imitating this construction, it is possible to define a much more general class of Hausdorff extensions of a locally bounded space with respect to a given boundedness, the so-called B-extensions. In this paper we study separation properties and metrizability of this kind of extension.      

A classification of CO spaces which are continuous images of compact ordered spaces

A topological space X is called a CO space, if every closed subset of X is homeomorphic to some clopen subset of X. Every ordinal with its order topology is a CO space. This work gives a complete classification of CO spaces which are continuous images of compact ordered spaces.     

On box products

We prove two theorems about box products. The first theorem says that the box product of countable spaces is pseudonormal, i.e. any two disjoint closed sets one of which is countable can be separated by open sets. The second theorem says that assuming CH a certain uncountable box product is normal (i.e. <_ω1?□_α<ω1X_α where each Xα is a compact metric space).     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

Hereditarily strongly cwH and other separation axioms

We explore the relation between two general kinds of separation properties. The first kind, which includes the classical separation properties of regularity and normality, has to do with expanding two disjoint closed sets, or dense subsets of each, to disjoint open sets. The second kind has to do with expanding discrete collections of points, or full-cardinality subcollections thereof, to disjoint or discrete collections of open sets. The properties of being collectionwise Hausdorff (cwH), of being strongly cwH, and of being wD(ℵ₁), fall into the second category. We study the effect on other separation properties if these properties are assumed to hold hereditarily. In the case of scattered spaces, we show that (a) the hereditarily cwH ones are α-normal and (b) a regular one is hereditarily strongly cwH iff it is hereditarily cwH and hereditarily β-normal. Examples are given in ZFC of (1) hereditarily strongly cwH spaces which fail to be regular, including one that also fails to be α-normal; (2) hereditarily strongly cwH regular spaces which fail to be normal and even, in one case, to be β-normal; (3) hereditarily cwH spaces which fail to be α-normal. We characterize those regular spaces X such that X×(ω+1) is hereditarily strongly cwH and, as a corollary, obtain a consistent example of a locally compact, first countable, hereditarily strongly cwH, non-normal space. The ZFC-independence of several statements involving the hereditarily wD(ℵ₁) property is established. In particular, several purely topological statements involving this property are shown to be equivalent to b=ω₁.     

Diagonalizing matrices over C(X)

A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)?M_n of continuous functions from X to $\text{M}$_n can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ? 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

A note on λ-compact spaces

We extend the well-known and important fact that “a topological space X is compact if and only if every ideal in C(X) is fixed”, to more general topological spaces. Some interesting consequences are also observed. In particular, the maximality of compact Hausdorff spaces with respect to the property of compactness is generalized and the topological spaces with this generalized property are characterized.     

On HCO spaces. An uncountable compactT 2 space,different from ℵ1+1, which is homeomorphic to each of its uncountable closed subspaces,which is homeomorphic to each of its uncountable closed subspaces

LetX be an Hausdorff space. We say thatX is a CO space, ifX is compact and every closed subspace ofX is homeomorphic to a clopen subspace ofX, andX is a hereditarily CO space (HCO space), if every closed subspace is a CO space. It is well-known that every well-ordered chain with a last element, endowed with the interval topology, is an HCO space, and every HCO space is scattered. In this paper, we show the following theorems: Theorem (R. Bonnet):

1. Every HCO space which is a continuous image of a compact totally disconnected interval space is homeomorphic to β+1 for some ordinal β.
2. Every HCO space of countable Cantor-Bendixson rank is homeomorphic to α+1 for some countable ordinal α.

Theorem (S. Shelah):Assume \(\diamondsuit _{\aleph _1 } \) . Then there is a HCO compact space X of Cantor-Bendixson rankω ₁} and of cardinality ?₁ such that:

1. X has only countably many isolated points,
2. Every closed subset of X is countable or co-countable,
3. Every countable closed subspace of X is homeomorphic to a clopen subspace, and every uncountable closed subspace of X is homeomorphic to X, and
4. X is retractive.

In particularX is a thin-tall compact space of countable spread, and is not a continuous image of a compact totally disconnected interval space. The question whether it is consistent with ZFC, that every HCO space is homeomorphic to an ordinal, is open.