nLab
absolute extensor

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

A topological space KK is called an absolute extensor if for

  1. XX any nice topological space

  2. AiXA \overset{i}{\hookrightarrow} X any closed subspace

  3. f:AKf\colon A \longrightarrow K any continuous function

there is an extension to a continuous function f˜:XK\tilde{f}:X\to K, i.e., such that f˜=fi\tilde{f}=f\circ i:

A f K i f˜ X \array{ A &\overset{f}{\longrightarrow}& K \\ {}^{\mathllap{i}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}} \\ X }

Here “nice topological space” is variously taken to mean metrizable topological space or at least normal topological space.

A variation of this concept, absolute neighborhood extension, only requires the extension to exist over a neighborhood of AA in XX.

Examples

The Tietze extension theorem implies that the real line \mathbb{R} equipped with its Euclidean space metric topology is an absolute extensor. It follows that so are the closed interval subspace [0,1][0,1] \subset \mathbb{R} and the circle S 1S^1.

Products of absolute extensors are absolute extensors, including the Hilbert cube.

The two point discrete space S 0S^0 as well as any sphere S nS^n is an absolute neighborhood extensor, but not an absolute extensor.

References

Last revised on May 24, 2017 at 02:58:41. See the history of this page for a list of all contributions to it.