Abstract
This chapter begins with a brief and informal description of the fixed point index, which is the book’s central concept. The fixed point index for Euclidean spaces assigns an integer to each continuous function from a compact subset of a Euclidean space to the Euclidean space that has no fixed points in its boundary. It is characterized by three properties: Normalization requires that the index of a constant function whose value is in the interior of the domain is \(+1\); Additivity requires that if finitely many compact subdomains are pairwise disjoint and contain the fixed points of the function in their interiors, then the index of the function is the sum of the indices of the restrictions of the functions to the subdomains; Continuity requires that the index agrees with the indices of nearby functions with the same domain. We then describe the history of the development of the topic, and give brief summaries of the other chapters.
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Notes
- 1.
In this book a topological space is compact if every open cover has a finite subcover. (In the Bourbaki tradition such a space is said to be quasicompact, and a compact space is one that is both quasicompact and Hausdorff.)
- 2.
Although it is not directly relevant, it makes sense to mention that F is lower hemicontinuous if, for each x and open \(V \subset Y\) such that \(F(x) \cap V \ne \emptyset \) there is a neighborhood U of x such that \(F(x') \cap V \ne \emptyset \) for all \(x' \in U\).
- 3.
Although the result is universally attributed to Brouwer, it seems that it had actually been proved earlier by Bohl (1904).
- 4.
We recall that a subset K of a topological space X is connected if there do not exist two disjoint open sets \(U_1\) and \(U_2\) with \(S \cap U_1 \ne \emptyset \ne S \cap U_2\) and \(K \subset U_1 \cup U_2\).
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McLennan, A. (2018). Introduction and Summary. In: Advanced Fixed Point Theory for Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0710-2_1
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DOI: https://doi.org/10.1007/978-981-13-0710-2_1
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Online ISBN: 978-981-13-0710-2
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