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

ČECH SYSTEM DOES NOT INDUCE APPROXIMATE SYSTEMS

Vlasta Matijević

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.

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

References:

1. S. Mardešić, Chainable continua and inverse limits, Glasnik Mat.-Fiz. Astronom. 14 (1959), 219-232.
   0126832

2. S. Mardešić, On covering dimension and inverse limits for compact spaces, Illinois J. Math. 4 (1960), 278-291.
   MathSciNet

3. S. Mardešić and L. R. Rubin, Approximate inverse systems of compacta and covering dimension, Pacific J. Math. 138 (1989), 129-144.
   MathSciNet

4. S. Mardešić and J. Segal, Shape Theory. The inverse system approach, North Holland, Amsterdam, 1982.
   MathSciNet

5. S. Mardešić and T. Watanabe, Approximate resolutions of spaces and mappings, Glas. Mat. Ser. III 24 (1989), 587-637.
   MathSciNet