Free spaces

Jian Song; E. Tymchatyn

Fundamenta Mathematicae (2000)

  • Volume: 163, Issue: 3, page 229-239
  • ISSN: 0016-2736

Abstract

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A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense G δ -subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will answer most of those questions. We prove that each cone is free. We introduce the notion of a locally free space and prove that a locally free ANR is free. It follows that every polyhedron is free. Hence, 1-dimensional Peano continua, Menger manifolds and many hereditarily unicoherent continua are free. We also give examples that show some limits to the extent of the class of free spaces.

How to cite

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Song, Jian, and Tymchatyn, E.. "Free spaces." Fundamenta Mathematicae 163.3 (2000): 229-239. <http://eudml.org/doc/212441>.

@article{Song2000,
abstract = {A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense $G_δ$-subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will answer most of those questions. We prove that each cone is free. We introduce the notion of a locally free space and prove that a locally free ANR is free. It follows that every polyhedron is free. Hence, 1-dimensional Peano continua, Menger manifolds and many hereditarily unicoherent continua are free. We also give examples that show some limits to the extent of the class of free spaces.},
author = {Song, Jian, Tymchatyn, E.},
journal = {Fundamenta Mathematicae},
keywords = {free space; hereditarily indecomposable continuum; polyhedron},
language = {eng},
number = {3},
pages = {229-239},
title = {Free spaces},
url = {http://eudml.org/doc/212441},
volume = {163},
year = {2000},
}

TY - JOUR
AU - Song, Jian
AU - Tymchatyn, E.
TI - Free spaces
JO - Fundamenta Mathematicae
PY - 2000
VL - 163
IS - 3
SP - 229
EP - 239
AB - A space Y is called a free space if for each compactum X the set of maps with hereditarily indecomposable fibers is a dense $G_δ$-subset of C(X,Y), the space of all continuous functions of X to Y. Levin proved that the interval I and the real line ℝ are free. Krasinkiewicz independently proved that each n-dimensional manifold M (n ≥ 1) is free and the product of any space with a free space is free. He also raised a number of questions about the extent of the class of free spaces. In this paper we will answer most of those questions. We prove that each cone is free. We introduce the notion of a locally free space and prove that a locally free ANR is free. It follows that every polyhedron is free. Hence, 1-dimensional Peano continua, Menger manifolds and many hereditarily unicoherent continua are free. We also give examples that show some limits to the extent of the class of free spaces.
LA - eng
KW - free space; hereditarily indecomposable continuum; polyhedron
UR - http://eudml.org/doc/212441
ER -

References

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  1. [B] M. Bestvina, Characterizing k-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 380 (1988). Zbl0645.54029
  2. [CKT] A. Chigogidze, K. Kawamura and E. D. Tymchatyn, Nöbeling spaces and pseudo-interiors of Menger compacta, Topology Appl. 68 (1996), 33-65. Zbl0869.57024
  3. [F1] J. B. Fugate, Small retractions of smooth dendroids onto trees, Fund. Math. 71 (1971), 255-262. Zbl0226.54030
  4. [F2] J. B. Fugate, Retracting fans onto finite fans, ibid., 113-125. Zbl0214.49701
  5. [HW] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, 1941. Zbl67.1092.03
  6. [K] J. Krasinkiewicz, On mappings with hereditarily indecomposable fibers, Bull. Polish Acad. Sci. Math. 44 (1996), 147-156. Zbl0867.54020
  7. [Ku] K. Kuratowski, Topology, Vols. I, II, Academic Press, New York, 1968. 
  8. [L] M. Levin, Bing maps and finite-dimensional maps, Fund. Math. 151 (1996), 47-52. Zbl0860.54028
  9. [M] J. van Mill, Infinite-Dimensional Topology, North-Holland, Amsterdam, 1989. Zbl0663.57001

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