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1
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A spectral characterization of skeletal maps

100%
Banakh T., Kucharski A., Martynenko M.
Open Mathematics
|
2013
|
tom 11
|
nr 1
161-169
EN
We prove that a map between two realcompact spaces is skeletal if and only if it is homeomorphic to the limit map of a skeletal morphism between ω-spectra with surjective limit projections.
2
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Skeletally Dugundji spaces

100%
Kucharski A., Plewik S., Valov V.
Open Mathematics
|
2013
|
tom 11
|
nr 11
1949-1959
EN
We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable inverse spectra, Uspekhi Mat. Nauk, 1976, 31(5), 191–226 (in Russian)] consisting of skeletal maps.
3
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On minimal Hausdorff and minimal Urysohn functions

64%
Cammaroto F., Catalioto A., Porter J.
Open Mathematics
|
2011
|
tom 9
|
nr 6
1242-1251
EN
In this article, we extend the work on minimal Hausdorff functions initiated by Cammaroto, Fedorchuk and Porter in a 1998 paper. Also, minimal Urysohn functions are introduced and developed. The properties of heredity and productivity are examined and developed for both minimal Hausdorff and minimal Urysohn functions.
4
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Reflecting topological properties in continuous images

64%
Tkachuk V.
Open Mathematics
|
2012
|
tom 10
|
nr 2
456-465
EN
Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ + if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ +. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ + for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ + for compact spaces. We present examples showing that separability, countable extent and normality do not reflect in continuous images of weight ω 1. Besides, under MA + ¬ CH, the Fréchet-Urysohn property does not reflect in continuous images of weight ω 1 even for compact spaces. An application of our techniques gives a solution of an open problem published by Ramírez-Páramo. If Jensen’s κ +-Axiom $$\left( {\diamondsuit _{\kappa ^ + } } \right)$$ holds for an infinite cardinal κ, then for an arbitrary space X with no G κ-points there exists a continuous surjective map f: X → Y such that w(Y) = κ + and Y has no G tk-points. We apply this result to solve a problem of Kalenda.
5
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On the extent of star countable spaces

52%
Alas O., Junqueira L., Mill J., Tkachuk V., Wilson R.
Open Mathematics
|
2011
|
tom 9
|
nr 3
603-615
EN
For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y ⊂ X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelöf spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelöf. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense σ-compact subspace can have arbitrary extent. It is proved that for any ω 1-monolithic compact space X, if C p(X)is star countable then it is Lindelöf.
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