Journal of Formalized Mathematics
Volume 3, 1991
University of Bialystok
Copyright (c) 1991 Association of Mizar Users

A Borsuk Theorem on Homotopy Types

Andrzej Trybulec
Warsaw University, Bialystok

Summary.

We present a Borsuk's theorem published first in [1] (compare also [2, pages 119-120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semi-continuous decompositions, retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved.

MML Identifier: BORSUK_1

The terminology and notation used in this paper have been introduced in the following articles [20] [8] [22] [11] [23] [5] [21] [19] [17] [13] [12] [3] [7] [16] [6] [15] [24] [10] [9] [14] [4] [18]

Contents (PDF format)

1. Preliminaries
2. Partitions
3. Topological Preliminaries
4. Cartesian Product of Topological Spaces
5. Partitions of Topological Spaces
6. Upper Semicontinuous Decompositions
7. Borsuk's Theorems on the Decomposition of Retracts

Bibliography

[1] Karol Borsuk. On the homotopy types of some decomposition spaces. \em Bull. Acad. Polon. Sci., (18):235--239, 1970.
[2] Karol Borsuk. \em Theory of Shape, volume 59 of \em Monografie Matematyczne. PWN, Warsaw, 1975.
[3] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[4] Czeslaw Bylinski. Basic functions and operations on functions. Journal of Formalized Mathematics, 1, 1989.
[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[6] Czeslaw Bylinski. Functions from a set to a set. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. Partial functions. Journal of Formalized Mathematics, 1, 1989.
[8] Czeslaw Bylinski. Some basic properties of sets. Journal of Formalized Mathematics, 1, 1989.
[9] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[10] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.
[11] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[12] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.
[13] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.
[14] Beata Padlewska. Locally connected spaces. Journal of Formalized Mathematics, 2, 1990.
[15] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[16] Konrad Raczkowski and Pawel Sadowski. Equivalence relations and classes of abstraction. Journal of Formalized Mathematics, 1, 1989.
[17] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.
[18] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.
[19] Andrzej Trybulec. Domains and their Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[20] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[21] Andrzej Trybulec. Tuples, projections and Cartesian products. Journal of Formalized Mathematics, 1, 1989.
[22] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[23] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[24] Miroslaw Wysocki and Agata Darmochwal. Subsets of topological spaces. Journal of Formalized Mathematics, 1, 1989.