Nice Hamel bases under the Covering Property Axiom

Krzysztof Ciesielski; Janusz Pawlikowski

doi:10.1023/b:amhu.0000049287.44877.2c

Nice Hamel bases under the Covering Property Axiom

Acta Mathematica Hungarica ◽

10.1023/b:amhu.0000049287.44877.2c ◽

2004 ◽

Vol 105 (3) ◽

pp. 197-213 ◽

Cited By ~ 6

Author(s):

Krzysztof Ciesielski ◽

Janusz Pawlikowski

Keyword(s):

Covering Property ◽

Covering Property Axiom

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Small Coverings with Smooth Functions under the Covering Property Axiom

Canadian Journal of Mathematics ◽

10.4153/cjm-2005-020-8 ◽

2005 ◽

Vol 57 (3) ◽

pp. 471-493 ◽

Cited By ~ 5

Author(s):

Krzysztof Ciesielski ◽

Janusz Pawlikowski

Keyword(s):

Borel Function ◽

Covering Property ◽

Smooth Functions ◽

Differentiable Functions ◽

Perfect Set ◽

Covering Property Axiom

AbstractIn the paper we formulate a Covering Property Axiom, CPAprism, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.(a) There exists a family ℱ of less than continuummany functions from ℝ to ℝ such that ℝ2 is covered by functions from ℱ, in the sense that for every 〈x, y〉 ∈ ℝ2 there exists an f ∈ ℱ such that either f (x) = y or f (y) = x.(b) For every Borel function f : ℝ → ℝ there exists a family ℱ of less than continuum many “” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of f.(c) For every n > 0 and a Dn function f: ℝ → ℝ there exists a family ℱ of less than continuum many Cn functions whose graphs cover the graph of f.We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiĭ.

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