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Continuous functions on limits of F-decomposable systems

Todor Manev

Abstract

We introduce the concept of F-decomposable systems, well-ordered inverse systems of Hausdorff compacta with fully closed bonding mappings. A continuous mapping between Hausdorff compacta is called fully closed if the intersection of the images of any two closed disjoint subsets is finite. We give a characterization of such systems in terms of a property of the continuous functions on their limit. When, moreover, the fibers of neighboring bonding mappings are metrizable, we call the limit of such a system an F_d-compact, a particular case of a Fedorchuk compact. The stated property allows us to obtain a locally uniformly rotund renorming on the space C(K), where K is an F_d-compact of countable spectral height.

Continuous functions on limits of F-decomposable systems

Abstract

We introduce the concept of F-decomposable systems, well-ordered inverse systems of Hausdorff compacta with fully closed bonding mappings. A continuous mapping between Hausdorff compacta is called fully closed if the intersection of the images of any two closed disjoint subsets is finite. We give a characterization of such systems in terms of a property of the continuous functions on their limit. When, moreover, the fibers of neighboring bonding mappings are metrizable, we call the limit of such a system an F_d-compact, a particular case of a Fedorchuk compact. The stated property allows us to obtain a locally uniformly rotund renorming on the space C(K), where K is an F_d-compact of countable spectral height.

Paper Structure

This paper contains 6 sections, 14 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. F-decomposable systems
  4. Construction
  5. Continuous functions on limits of inverse systems
  6. $\mathrm{F}_d$ compacta and LUR renormability

Key Result

Proposition 2.1

Let $X$ and $Y$ be Hausdorff compact spaces and $\varphi: X \to Y$ a continuous surjection. Let $\varphi^0$ denote the natural embedding of $C(Y)$ into $C(X)$, that is, $(\varphi^0 g)(x) := g(\varphi(x))$, for $g \in C(Y)$. Then, whenever $f \in C(X)$, there exists a function $g \in C(Y)$ such that

Figures (1)

  • Figure 1: A diagram of a three-level construction. Here $M_1 \subset Z$ and $M_2 \subset Y$.

Theorems & Definitions (26)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Proposition 2.1: see e.g. holmes_opt
  • proof
  • Remark
  • Proposition 2.3: fedFC
  • Proposition 2.4: fedFC
  • Corollary 2.5: fedFC
  • Proposition 2.6: fedFC
  • ...and 16 more