## A Borsuk theorem on homotopy types (1991)

Venue: | Journal of Formalized Mathematics |

Citations: | 99 - 7 self |

### BibTeX

@ARTICLE{Trybulec91aborsuk,

author = {Andrzej Trybulec},

title = {A Borsuk theorem on homotopy types},

journal = {Journal of Formalized Mathematics},

year = {1991},

pages = {3--1}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

1280 | Tarski Grothendieck set theory
- Trybulec
- 1990
(Show Context)
Citation Context ..., retracts, strong deformation retract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles =-=[20]-=-, [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use... |

1269 | Functions and their basic properties
- Byliński
- 1990
(Show Context)
Citation Context ...n retract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], =-=[5]-=-, [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e... |

1200 | Properties of subsets
- Trybulec
- 1990
(Show Context)
Citation Context ... strong deformation retract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], =-=[22]-=-, [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the follow... |

1025 | Functions from a set to a set
- Byliński
- 1990
(Show Context)
Citation Context ...sary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], =-=[6]-=-, [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subs... |

999 | Relations and their basic properties
- Woronowicz
- 1990
(Show Context)
Citation Context ...rmation retract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], =-=[23]-=-, [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following conventi... |

735 | Some basic properties of sets
- Byliński
- 1990
(Show Context)
Citation Context ...acts, strong deformation retract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], =-=[8]-=-, [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the ... |

444 | Partial functions
- Byliński
- 1990
(Show Context)
Citation Context ...t are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], =-=[7]-=-, [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and ... |

317 | Finite sets
- Darmochwał
- 1990
(Show Context)
Citation Context ...g deformation retract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], =-=[11]-=-, [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following co... |

304 | Families of sets
- Padlewska
- 1990
(Show Context)
Citation Context ...y those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], =-=[13]-=-, [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, ... |

297 | Domains and their Cartesian products
- Trybulec
- 1990
(Show Context)
Citation Context ...However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], =-=[19]-=-, [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , ... |

271 | Tuples, projections and Cartesian products
- Trybulec
- 1990
(Show Context)
Citation Context ...ract. However, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], =-=[21]-=-, [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X... |

264 | Binary operations applied to functions
- Trybulec
- 1990
(Show Context)
Citation Context ...ntifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and =-=[18]-=- provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subset of X. We now state a number of propositio... |

157 | The Euclidean space
- Darmochwał
- 1991
(Show Context)
Citation Context ...been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], =-=[9]-=-, [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subset of X. We now state a... |

146 | Topological properties of subsets in real numbers
- Raczkowski, Sadowski
- 1990
(Show Context)
Citation Context ...r, only those facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], =-=[17]-=-, [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2... |

114 | Basic functions and operations on functions
- Byliński
- 1990
(Show Context)
Citation Context .... MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], [14], =-=[4]-=-, and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subset of X. We now state a number of ... |

112 | Paracompact and metrizable spaces
- Borys
- 1991
(Show Context)
Citation Context ...s that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], =-=[3]-=-, [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets... |

104 |
Equivalence relations and classes of abstraction
- Raczkowski, Sadowski
- 1990
(Show Context)
Citation Context ... necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], =-=[16]-=-, [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a... |

92 | Connected spaces
- Padlewska
- 1990
(Show Context)
Citation Context ...proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], [10], [9], =-=[14]-=-, [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subset of X. We now state a numbe... |

73 |
Mariusz Startek. Metric spaces
- Kanas, Lecko
- 1990
(Show Context)
Citation Context ...e facts that are necessary in the proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], =-=[12]-=-, [3], [7], [16], [6], [15], [24], [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are... |

69 | Families of subsets, subspaces and mappings in topological spaces
- Darmochwał
- 1990
(Show Context)
Citation Context ... have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], [24], =-=[10]-=-, [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subset of X. We now st... |

44 |
Subsets of topological spaces
- Wysocki, Darmochwal
- 1989
(Show Context)
Citation Context ... proof have been proved. MML Identifier: BORSUK_1. WWW: http://mizar.org/JFM/Vol3/borsuk_1.html The articles [20], [8], [22], [11], [23], [5], [21], [19], [17], [13], [12], [3], [7], [16], [6], [15], =-=[24]-=-, [10], [9], [14], [4], and [18] provide the notation and terminology for this paper. 1. PRELIMINARIES We use the following convention: e, u, X, Y , X1, X2, Y1, Y2 are sets and A is a subset of X. We ... |

4 |
On the homotopy type of some decomposition spaces
- Borsuk
- 1970
(Show Context)
Citation Context ... Published 2003 Inst. of Computer Science, Univ. of Białystok A Borsuk Theorem on Homotopy Types Andrzej Trybulec Warsaw University Białystok 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... |

2 | Theory of Shape. Volume 59 of Monografie Matematyczne - Borsuk - 1975 |