σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

Taras Banakh

doi:10.2478/tmmp-2020-0014

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σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

Tatra Mountains Mathematical Publications

Volume 76 (2020): Issue 1 (December 2020)

By: Taras Banakh

Open Access

|Nov 2020

Abstract

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

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Articles in this issue

DOI: https://doi.org/10.2478/tmmp-2020-0014 | Journal eISSN: 1338-9750 | Journal ISSN: 1210-3195

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Language: English

Page range: 1 - 10

Submitted on: Sep 15, 2019

Published on: Nov 4, 2020

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 1 issue per year

Keywords:

σ-continuous function,

¯σ-continuous function,

cardinal characteristic of the continuum

Related subjects:

Mathematics,

General mathematics

© 2020 Taras Banakh, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 76 (2020): Issue 1 (December 2020)