Original scientific paper

https://doi.org/10.21857/9e31lh4nrm

Čech system does not induce approximate systems

Vlasta Matijević ; Department of Mathematics, Faculty of Science, University of Split, 21 000 Split, Croatia

Abstract

With every topological space X is associated its Čech system C(X)=(|N(U)|, [pUV], Cov(X)). It is well-known that the Čech system C(X) of X is an inverse system in the homotopy category HPol whose objects are polyhedra and morphisms are homotopy classes of continuous maps between polyhedra. We consider the following question posed by S. Mardešić. For a given Čech system (|N(U)|, [pUV], Cov(X)) of a space X, is it possible to select a member qUV ∈ [pUV] in each homotopy class [pUV] in such a way that the obtained system (|N(U)|, [qUV], Cov(X)) is an approximate system? We answer the question in the negative by proving that for each Hausdorff arc-like continuum X any such system (|N(U)|, [qUV], Cov(X)) is not an approximate system.

Keywords

Inverse system; Čech system; approximate system; polyhedro; nerve of a covering; arc-like space; chainable space

Hrčak ID:

186436

URI

https://hrcak.srce.hr/186436

Publication date:

13.9.2017.

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