On a finite type multivalued shape. (English) Zbl 1520.54011
The author first provides an overview of the development of shape theory. Basics related to different approaches to shape are given. Special emphasis is placed on the so called Main Construction of the multivalued maps approach to shape theory of metric compacta.
In the main part of the article the author introduces the notion of an approximative map of finite type. It is shown (Theorem 1) that for every approximative map \(\left\lbrace f_n\right\rbrace_{n\in\mathbb{N}}\) between compact metric spaces \(X\) and \(Y\) there exists an approximative map \(\left\lbrace f'_n\right\rbrace_{n\in\mathbb{N}}\) of finite type from \(X\) to \(Y\) which is homotopic to \(\left\lbrace f_n\right\rbrace_{n\in\mathbb{N}}\). This Theorem, together with two additional Propositions, proves that the Main Construction captures the shape properties of the space in which it is done and that it is able to do so using only finite subsets of the space.
In the main part of the article the author introduces the notion of an approximative map of finite type. It is shown (Theorem 1) that for every approximative map \(\left\lbrace f_n\right\rbrace_{n\in\mathbb{N}}\) between compact metric spaces \(X\) and \(Y\) there exists an approximative map \(\left\lbrace f'_n\right\rbrace_{n\in\mathbb{N}}\) of finite type from \(X\) to \(Y\) which is homotopic to \(\left\lbrace f_n\right\rbrace_{n\in\mathbb{N}}\). This Theorem, together with two additional Propositions, proves that the Main Construction captures the shape properties of the space in which it is done and that it is able to do so using only finite subsets of the space.
Reviewer: Ivan Jelić (Split)
MSC:
| 54C56 | Shape theory in general topology |
| 54C60 | Set-valued maps in general topology |
| 54D10 | Lower separation axioms (\(T_0\)–\(T_3\), etc.) |
| 55P55 | Shape theory |
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