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On maps preserving connectedness and/or compactness

István Juhász, Jan van Mill (2018)

Commentationes Mathematicae Universitatis Carolinae

We call a function f : X → Y P-preserving if, for every subspace A ⊂ X with property P, its image f ( A ) also has property P. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions such a map is continuous, has a long history. Our main result is that any nontrivial product function, i.e. one having at least two nonconstant factors, that has connected domain, T 1 range, and is connectedness-preserving...

On Mazurkiewicz sets

Marta N. Charatonik, Włodzimierz J. Charatonik (2000)

Commentationes Mathematicae Universitatis Carolinae

A Mazurkiewicz set M is a subset of a plane with the property that each straight line intersects M in exactly two points. We modify the original construction to obtain a Mazurkiewicz set which does not contain vertices of an equilateral triangle or a square. This answers some questions by L.D. Loveland and S.M. Loveland. We also use similar methods to construct a bounded noncompact, nonconnected generalized Mazurkiewicz set.

On multifunctions with closed graphs

D. Holý (2001)

Mathematica Bohemica

The set of points of upper semicontinuity of multi-valued mappings with a closed graph is studied. A topology on the space of multi-valued mappings with a closed graph is introduced.

On n -thin dense sets in powers of topological spaces

Adam Bartoš (2016)

Commentationes Mathematicae Universitatis Carolinae

A subset of a product of topological spaces is called n -thin if every its two distinct points differ in at least n coordinates. We generalize a construction of Gruenhage, Natkaniec, and Piotrowski, and obtain, under CH, a countable T 3 space X without isolated points such that X n contains an n -thin dense subset, but X n + 1 does not contain any n -thin dense subset. We also observe that part of the construction can be carried out under MA.

On powers of Lindelöf spaces

Isaac Gorelic (1994)

Commentationes Mathematicae Universitatis Carolinae

We present a forcing construction of a Hausdorff zero-dimensional Lindelöf space X whose square X 2 is again Lindelöf but its cube X 3 has a closed discrete subspace of size 𝔠 + , hence the Lindelöf degree L ( X 3 ) = 𝔠 + . In our model the Continuum Hypothesis holds true. After that we give a description of a forcing notion to get a space X such that L ( X n ) = ℵ 0 for all positive integers n , but L ( X ℵ 0 ) = 𝔠 + = ℵ 2 .

On projectively quotient functors

T. F. Zhuraev (2001)

Commentationes Mathematicae Universitatis Carolinae

We introduce notions of projectively quotient, open, and closed functors. We give sufficient conditions for a functor to be projectively quotient. In particular, any finitary normal functor is projectively quotient. We prove that the sufficient conditions obtained are necessary for an arbitrary subfunctor ℱ of the functor 𝒫 of probability measures. At the same time, any “good” functor is neither projectively open nor projectively closed.

Currently displaying 81 – 100 of 187