nLab quasi-finite CW-complex




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

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


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



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).


  • 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

Last revised on May 23, 2017 at 18:16:33. See the history of this page for a list of all contributions to it.