**Abstract.** The notion of shape fibration between compact metric spaces
was introduced by S. Mardesic and T. B. Rushing. Mardesic extended the notion to
arbitrary topological spaces. A shape fibration
*f* : *X* → *Y***p**, **q**, **f**) of the map *f*,
where **p** : *X* → **X****q** : *Y* → **Y***X* and *Y*, respectively, and the approximate homotopy lifting property
for the system map **f** : **X** → **Y**.*f* : *X* → *Y***p**, **q**, **f**), if polyhedral resolutions
**p** : *X* → **X****q** : *Y* → **Y****f** : **X** → **Y****p**,**q**,**f**) is a resolution of *f*.
To overcome this deficiency, T. Watanabe introduced the notion of approximate resolution.
An approximate resolution of a map
*f* : *X* → *Y***p** : *X* → X**q** : *Y* → Y*X* and *Y*, respectively, and an approximate map
**f** : X →
Y.

**2000 Mathematics Subject Classification.**
54C56, 55P55.

**Key words and phrases.** Approximate inverse system, shape fibration, approximate shape, fibration category.

DOI: 10.3336/gm.44.1.14

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MathSciNet