<abstract xmlns="http://www.w3.org/1999/xhtml">A function <italic>f</italic> : <italic>X</italic> → <italic>Y</italic> between topological spaces is called <italic>σ-continuous</italic> (resp. <italic>̄σ-continuous</italic>) if there exists a (closed) cover {<italic>Xn</italic>}<italic>n</italic>∈<italic>ω</italic> of <italic>X</italic> such that for every <italic>n</italic> ∈ <italic>ω</italic> the restriction <italic>f</italic> ↾ <italic>Xn</italic> is continuous. By 𝔠<italic> σ</italic> (resp. 𝔠¯σ)we denote the largest cardinal <italic>κ</italic> ≤ 𝔠 such that every function <italic>f</italic> : <italic>X</italic> → ℝ defined on a subset <italic>X</italic> ⊂ ℝ of cardinality |<italic>X</italic>| <italic>&lt;κ</italic> is <italic>σ</italic>-continuous (resp. ¯<italic>σ</italic>-continuous). It is clear that <italic>ω</italic>1 ≤ 𝔠¯<italic>σ</italic> ≤ 𝔠 <italic>σ</italic> ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯<italic>σ</italic> =min{𝔠<italic> σ</italic>, 𝔟, 𝔮 }≤ 𝔠 <italic>σ</italic> ≤ min{non(ℳ), non(𝒩)}.</abstract>

A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn is continuous. By 𝔠 σ (resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω1 ≤ 𝔠¯σ ≤ 𝔠 σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯σ =min{𝔠 σ, 𝔟, 𝔮 }≤ 𝔠 σ ≤ min{non(ℳ), non(𝒩)}.

<italic>σ</italic>-Continuous Functions and Related Cardinal Characteristics of the Continuum

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

-Continuous Functions and Related Cardinal Characteristics of the Continuum

A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every...

σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

Published Online: Nov 04, 2020

Page range: 1 - 10

Received: Sep 15, 2019

DOI: https://doi.org/10.2478/tmmp-2020-0014

Keywords
-continuous function, ¯-continuous function, cardinal characteristic of the continuum

© 2020 Taras Banakh, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

Published Online: Nov 04, 2020

Page range: 1 - 10

Received: Sep 15, 2019

DOI: https://doi.org/10.2478/tmmp-2020-0014

Keywords-continuous function, ¯-continuous function, cardinal characteristic of the continuum

© 2020 Taras Banakh, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
-continuous function, ¯-continuous function, cardinal characteristic of the continuum