#### Rad HAZU, Matematičke znanosti, Vol. 21 (2017), 169-177.

### ČECH SYSTEM DOES NOT INDUCE APPROXIMATE SYSTEMS

### Vlasta Matijević

Department of Mathematics, Faculty of Science, University of Split, 21 000 Split, Croatia

*e-mail:* `vlasta@pmfst.hr`

**Abstract.** With every topological space *X* is associated its
Čech system
**C**(*X*)=(|*N*(U)|,
[*p*_{UV}], *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)|,
[*p*_{UV}], *Cov*(*X*))
of a space *X*, is it possible to select a member
*q*_{UV} ∈
[*p*_{UV}]
in each homotopy class
[*p*_{UV}]
in such
a way that the obtained system
(|*N*(U)|,
[*q*_{UV}], *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)|,
[*q*_{UV}], *Cov*(*X*))
is not an approximate system.

**2010 Mathematics Subject Classification.**
54B35, 18B30, 54D30.

**Key words and phrases.** Inverse system, Čech system, approximate system, polyhedron,
nerve of a covering, arc-like space, chainable space.

**Full text (PDF)** (free access)
DOI: http://doi.org/10.21857/9e31lh4nrm

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0126832

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MathSciNet

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MathSciNet

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