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On two consequences of CH established by Sierpinski. II

Roman Pol, Piotr Zakrzewski

Abstract

We continue a study of the relations between two consequences of the Continuum Hypothesis discovered by Waclaw Sierpinski, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.

On two consequences of CH established by Sierpinski. II

Abstract

We continue a study of the relations between two consequences of the Continuum Hypothesis discovered by Waclaw Sierpinski, concerning uniform continuity of continuous functions and uniform convergence of sequences of real-valued functions, defined on subsets of the real line of cardinality continuum.
Paper Structure (7 sections, 13 theorems, 9 equations)

This paper contains 7 sections, 13 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Mappings into non-compact spaces and $C_9$
  3. Mappings into compact spaces and $C_8$
  4. Constructing $C_8$-like examples from $\mathscr{K}$-Lusin sets
  5. Zero-dimensional spaces
  6. Infinite-dimensional spaces
  7. A characterization of complete metrizability

Key Result

Proposition 2.1

If there exist a separable metric space $(X,d_X)$ of cardinality $\lambda$, a metric space $(Y,d_Y)$, and a continuous function on $X$ with range in $Y$, which is not uniformly continuous on any subset of $X$ of cardinality $\kappa$, then there exists a $\kappa$-K-Lusin set of cardinality $\lambda$

Theorems & Definitions (27)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Proposition 4.1
  • ...and 17 more