On a generalization of perfectly normal spaces

The notion of quasi-perfect normality which generalizes the notion of usual perfect normality is introduced and the behavior of the dimension functions in this class of spaces is investigated.     

The evaluation identification in function spaces

A study is made of the natural function which maps each point x of a space X to the evaluation function e_x:Y^x→Y defined by $\text{e}{}_{\mathrm{x}}\text{(?)=?(x)}$. A consequence of the results is that βX and υX can both be considered as subspaces of spaces of continous functions from appropriate domain spaces into I or R, respectively.     

On the topology of graph picture spaces

We study the space of pictures of a graph G in complex projective d-space. The main result is that the homology groups (with integer coefficients) of are completely determined by the Tutte polynomial of G. One application is a criterion in terms of the Tutte polynomial for independence in the d-parallel matroids studied in combinatorial rigidity theory. For certain special graphs called orchards, the picture space is smooth and has the structure of an iterated projective bundle. We give a Borel presentation of the cohomology ring of the picture space of an orchard, and use this presentation to develop an analogue of the classical Schubert calculus.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

On sieve-complete and compact-like spaces

It is shown that a completely regular space X is sieve-complete (or, equivalenty, X is the open image of a paracompact ?ech-complete space) iff βX?X is compact-like, i.e., Player I has a winning strategy in the topological game G(C, βX?X) of [13].     

Linearly ordered covers,normality and paracompactness

It is shown that a regular space is collectionwise normal and countably paracompact if every open cover has an open, order cushioned refinement. A sufficient condition for paracompactness, in terms of certain order locally finite covers, is given, and is applied to the problem of the paracompactness of product spaces.     

On strongly replete lattices,support of A measure,and the wallman remainder

LetX be an abstract set andL a lattice of subsets ofX. To eachL-regular measure on the algebra generated byL, there are associated two measures on appropriate algebras of the Wallman space. In terms of these measures, we can obtain characterization for-smoothness,-smoothness, and tightness of the original measure. In particular, tight regular measures and their properties are investigated.     

Small Valdivia compact spaces

We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight?ℵ₁ is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight?ℵ₁ is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.     

Boundedness properties in function spaces

Some boundedness properties of function spaces (considered as topo- logical groups) are studied.     

Non-regular power homogeneous spaces

We show that the cardinality of any space X with Δ-power homogeneous semiregularization that is either Urysohn or quasiregular is bounded by ²c(X)πχ(X). This improves a result of G.J. Ridderbos who showed this bound holds for Δ-power homogeneous regular spaces. By introducing the notion of a local πθ-base, we show that this bound can be further sharpened. We also show that no H-closed extremally disconnected space is power homogeneous. This is a variation of a result of K. Kunen who showed that no compact F-space is power homogeneous.     

Local n-connectivity of decomposition spaces

A corollary of the main result of this paper is the following Theorem. Suppose f: X → Y is a closed surjection of metrizable spaces whose point inverses are LC^{n + 1}-divisors (n ? 1). If Y is complete and f is homology n-stable, then Y is LC^{n + 1}provided X is LC^{n + 1}.Intuitively, f is homology n-stable if the ?ech homology groups of its point inverses are locally constant up to dimension n. In addition, we obtain sufficient conditions for the Freudenthal compactification to be LCⁿ.     

Products of Compacta with Plyhedra and Topological Spaces in the Shape Category

In the literature there exist examples of separable metric spaces X,Y whose Cartesian product X × Y is not a product in the shape category Sh(Top). It is an open question whether, for X a compact Hausdorff space, X × Y is a product in Sh(Top), for every topological spaces Y. The main result of the paper asserts that the answer is positive provided X × P is a product in Sh(Top), for every polyhedron P.     

Compact spaces generated by retractions

We study compact spaces which are obtained from metric compacta by iterating the operation of inverse limit of continuous sequences of retractions. This class, denoted by R, has been introduced in [M. Burke, W. Kubi?, S. Todor?evi?, Kadec norms on spaces of continuous functions, http://arxiv.org/abs/math.FA/0312013]. Allowing continuous images in the definition of class R, one obtains a strictly larger class, which we denote by RC. We show that every space in class RC is either Corson compact or else contains a copy of the ordinal segment ω₁+1. This improves a result of Kalenda from [O. Kalenda, Embedding of the ordinal segment [0,ω₁] into continuous images of Valdivia compacta, Comment. Math. Univ. Carolin. 40 (4) (1999) 777-783], where the same was proved for the class of continuous images of Valdivia compacta. We prove that spaces in class R do not contain cutting P-points (see the definition below), which provides a tool for finding spaces in RC?R. Finally, we study linearly ordered spaces in class RC. We prove that scattered linearly ordered compacta belong to RC and we characterize those ones which belong to R. We show that there are only 5 types (up to order isomorphism) of connected linearly ordered spaces in class R and all of them are Valdivia compact. Finally, we find a universal pre-image for the class of all linearly ordered Valdivia compacta.     

Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR ^2d , which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol lies in the modulation spaceM _,1 (R ^2d ), then the corresponding pseudodifferential operator is bounded onL ²(R ^d ) and, more generally, on the modulation spacesM _p,p (R ^d ) for 1p. If lies in the modulation spaceM _2,2 ^s (R ^2d )=L _s ^/2 (R ^2d )H ^s (R ^2d ), i.e., the intersection of a weightedL ²-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.     

On some questions about KC and related spaces

Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples:

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a KC space which cannot be embedded in any compact KC space;

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a countable KC space which does not admit any coarser compact KC topology;

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a minimal Hausdorff space which is not a k-space.

We also give an example of a compact KC space such that every nonempty open subset of it is dense, even if, as pointed out to us by the referee, a completely different construction carried out by E.K. van Douwen in 1993 leads to a space with the same properties.     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

On projections and limit mappings of inverse systems of compact spaces

A compactificaton αX of a completely regular space X is “determined” by a subset F of C¹(X) if αX is the smallest compactificaton of X to which each element of F extends, and is “generated” by F if the evaluation map $\text{e}{}_{\mathrm{F}}\text{:X →}\text{R}{}^{\mathrm{n}}\text{,n = |F|}$, is an embedding and $\text{αX =}\text{e}{}_{\mathrm{F}}\text{(X)}$. Evidently, if F either determines or generates αX, then every elements of F has an extension to αX; whenever F satisfies this latter condition, the set of all such extensions is denoted F^α.A major results of our previous paper is that F determines αX if and only if F^α separates points of αX ? X. A major result of the present paper is that F generates αX if and only if F^α separates points of αX.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.