Reflecting topological properties in continuous images

• Autorzy: Vladimir Tkachuk
• Języki publikacji: EN

Abstrakty

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 54C25: Embedding
• 54C10: Special maps on topological spaces (open, closed, perfect, etc.)
• 54H11: Topological groups
• 54D25: `` P -minimal'' and `` P -closed'' spaces

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 456-465

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Vladimir Tkachuk

• Universidad Autónoma Metropolitana

Bibliografia

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• [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
• [13] Tkachuk V.V., Spaces that are projective with respect to classes of mappings, Trans. Moscow Math. Soc., 1988, 139–156
• [14] Tkachuk V.V., A short proof of a classical result of M.G. Tkachenko, Topology Proc., 2001/02, 26(2), 851–856
• [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

Identyfikatory

DOI

10.2478/s11533-011-0141-9