nLab
quasi-finite CW-complex

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

Definition

A countable CW-complex KK is quasi-finite if for any finite subcomplex MKM\subset K, there is (possibly larger) finite subcomplex e(M)Ke(M)\subset K, such that for every separable metric space XX satisfying

  • (KK is an absolute extensor of XX:) for every closed subspace AXA\subset X and a function f:AKf:A\to K there is an extension f˜:XK\tilde{f}:X\to K (i.e. f˜=fi\tilde{f}=f\circ i, where i:AXi:A\hookrightarrow X is the closed embedding)

one has a similar property

  • for every closed subspace AXA\subset X and a function f:AMf:A\to M there is an extension g:Xe(M)g:X\to e(M) (i.e. gi=fg\circ i = f).

There is a characterization: a coutable CW-complex KK is quasi-finite iff for all separable metric spaces XX, if KK is an absolute extensor of XX implies then it is an absolute extensor of its Stone-Čech compactification β(X)\beta(X) as well.

In fact, the original definition asks that one has a function e:Me(M)e:M\to e(M) (the same under the axiom of choice).

References

  • A.Karasev, On two problems in extension theory, arXiv:math.GT/0312269

  • M.Cencelj, J.Dydak, J.Smrekar, A.Vavpetic, Ž.Virk, Algebraic properties of quasi-finite complexes, Fund. Math. 197 (2007), 67-80 math/0509582

Revised on May 23, 2017 14:16:33 by Urs Schreiber (92.218.150.85)