# Reflecting topological properties in continuous images

Open Mathematics (2012)

- Volume: 10, Issue: 2, page 456-465
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topVladimir Tkachuk. "Reflecting topological properties in continuous images." Open Mathematics 10.2 (2012): 456-465. <http://eudml.org/doc/269228>.

@article{VladimirTkachuk2012,

abstract = {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.},

author = {Vladimir Tkachuk},

journal = {Open Mathematics},

keywords = {Continuous images; Reflected properties; Small weight; Jensen’s Axiom; Diamond principle; Space with no G
K-points; Spread; Extent; Souslin number; Weight; Network weight; Corson space; Character; π-character; continuous images; reflected properties; small weight; character; pseudocharacter; network weight},

language = {eng},

number = {2},

pages = {456-465},

title = {Reflecting topological properties in continuous images},

url = {http://eudml.org/doc/269228},

volume = {10},

year = {2012},

}

TY - JOUR

AU - Vladimir Tkachuk

TI - Reflecting topological properties in continuous images

JO - Open Mathematics

PY - 2012

VL - 10

IS - 2

SP - 456

EP - 465

AB - 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.

LA - eng

KW - Continuous images; Reflected properties; Small weight; Jensen’s Axiom; Diamond principle; Space with no G
K-points; Spread; Extent; Souslin number; Weight; Network weight; Corson space; Character; π-character; continuous images; reflected properties; small weight; character; pseudocharacter; network weight

UR - http://eudml.org/doc/269228

ER -

## References

top- [1] Alas O.T., Tkachuk V.V., Wilson R.G., Closures of discrete sets often reflect global properties, Topology Proc., 2000, 25(Spring), 27–44 Zbl1002.54021
- [2] Arkhangel’skił A.V., Continuous mappings, factorization theorems and spaces of functions, Trudy Moskov. Mat. Obshch., 1984, 47, 3–21 (in Russian)
- [3] Arkhangel’skił A.V., Topological Function Spaces, Kluwer, Dordrecht, 1992 http://dx.doi.org/10.1007/978-94-011-2598-7
- [4] Dow A., An empty class of nonmetric spaces, Proc. Amer. Math. Soc., 1988, 104(3), 999–1001 http://dx.doi.org/10.1090/S0002-9939-1988-0964886-9 Zbl0692.54018
- [5] Engelking R., General Topology, Mathematical Monographs, 60, PWN, Warsaw, 1977
- [6] Gul’ko S.P., Properties of sets that lie in Σ-products, Dokl. Akad. Nauk SSSR, 1977, 237(3), 505–508 (in Russian)
- [7] Hajnal A., Juhász I., Having a small weight is determined by the small subspaces, Proc. Amer. Math. Soc., 1980, 79(4), 657–658 http://dx.doi.org/10.1090/S0002-9939-1980-0572322-2 Zbl0432.54003
- [8] Juhász I., Consistency results in topology, In: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, 503–522 http://dx.doi.org/10.1016/S0049-237X(08)71112-1
- [9] Juhász I., Cardinal Functions in Topology - Ten Years Later, Math. Centre Tracts, 123, Mathematisch Centrum, Amsterdam, 1980 Zbl0479.54001
- [10] Kalenda O.F.K., Note on countable unions of Corson countably compact spaces, Comment. Math. Univ. Carolin., 2004, 45(3), 499–507 Zbl1098.54020
- [11] Ramírez-Páramo A., A reflection theorem for i-weight, Topology Proc., 2004, 28(1), 277–281
- [12] Tkachenko M.G., Continuous mappings onto spaces of smaller weight, Moscow Univ. Math. Bull., 1980, 35(2), 41–44 Zbl0459.54003
- [13] Tkachuk V.V., Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc., 1988, 139–156 Zbl0662.54007
- [14] Tkachuk V.V., A short proof of a classical result of M.G. Tkachenko, Topology Proc., 2001/02, 26(2), 851–856 Zbl1083.54017
- [15] Tkachuk V.V., A C p-Theory Problem Book, Springer, New York-Dordrecht-Heidelberg-London, 2011 http://dx.doi.org/10.1007/978-1-4419-7442-6

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.