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On universality of countable and weak products of sigma hereditarily disconnected spaces

Taras Banakh, Robert Cauty (2001)

Fundamenta Mathematicae

Suppose a metrizable separable space Y is sigma hereditarily disconnected, i.e., it is a countable union of hereditarily disconnected subspaces. We prove that the countable power X ω of any subspace X ⊂ Y is not universal for the class ₂ of absolute G δ σ -sets; moreover, if Y is an absolute F σ δ -set, then X ω contains no closed topological copy of the Nagata space = W(I,ℙ); if Y is an absolute G δ -set, then X ω contains no closed copy of the Smirnov space σ = W(I,0). On the other hand, the countable power X ω of...

On universality of finite powers of locally path-connected meager spaces

Taras Banakh, Robert Cauty (2005)

Colloquium Mathematicae

It is shown that for every integer n the (2n+1)th power of any locally path-connected metrizable space of the first Baire category is 𝓐₁[n]-universal, i.e., contains a closed topological copy of each at most n-dimensional metrizable σ-compact space. Also a one-dimensional σ-compact absolute retract X is found such that the power X^{n+1} is 𝓐₁[n]-universal for every n.

On weakly infinite-dimensional subspuees

P. Borst (1992)

Fundamenta Mathematicae

We will construct weakly infinite-dimensional (in the sense of Y. Smirnov) spaces X and Y such that Y contains X topologically and d i m Y = ω 0 and d i m X = ω 0 + 1 . Consequently, the subspace theorem does not hold for the transfinite dimension dim for weakly infinite-dimensional spaces.

On weakly monotonically monolithic spaces

Liang-Xue Peng (2010)

Commentationes Mathematicae Universitatis Carolinae

In this note, we introduce the concept of weakly monotonically monolithic spaces, and show that every weakly monotonically monolithic space is a D -space. Thus most known conclusions on D -spaces can be obtained by this conclusion. As a corollary, we have that if a regular space X is sequential and has a point-countable w c s * -network then X is a D -space.

Open maps between Knaster continua

Carl Eberhart, J. Fugate, Shannon Schumann (1999)

Fundamenta Mathematicae

We investigate the set of open maps from one Knaster continuum to another. A structure theorem for the semigroup of open induced maps on a Knaster continuum is obtained. Homeomorphisms which are not induced are constructed, and it is shown that the induced open maps are dense in the space of open maps between two Knaster continua. Results about the structure of the semigroup of open maps on a Knaster continuum are obtained and two questions about the structure are posed.

Open maps having the Bula property

Valentin Gutev, Vesko Valov (2009)

Fundamenta Mathematicae

An open continuous map f from a space X onto a paracompact C-space Y admits two disjoint closed sets F₀,F₁ ⊂ X with f(F₀) = Y = f(F₁), provided all fibers of f are infinite and C*-embedded in X. Applications are given to the existence of "disjoint" usco multiselections of set-valued l.s.c. mappings defined on paracompact C-spaces, and to special type of factorizations of open continuous maps from metrizable spaces onto paracompact C-spaces. This settles several open questions.

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