Inessentiality with respect to subspaces

Michael Levin

Fundamenta Mathematicae (1995)

  • Volume: 147, Issue: 1, page 93-68
  • ISSN: 0016-2736

Abstract

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Let X be a compactum and let A = ( A i , B i ) : i = 1 , 2 , . . . be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed F i separating A i and B i the intersection ( ∩ F i ) ∩ Y is not empty. So A is inessential on Y if there exist closed F i separating A i and B i such that ∩ F i does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on which A is inessential and for every positive-dimensional Y ∩ X ╲ G there exists an infinite subfamily B ∩ A which is essential on Y.  >This theorem and its generalization provide a new approach for constructing hereditarily infinite-dimensional compacta not containing subspaces of positive dimension which are weakly infinite-dimensional or C-spaces.

How to cite

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Levin, Michael. "Inessentiality with respect to subspaces." Fundamenta Mathematicae 147.1 (1995): 93-68. <http://eudml.org/doc/212077>.

@article{Levin1995,
abstract = {Let X be a compactum and let $A=\{(A_i,B_i):i=1,2,...\}$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_i$ separating $A_i$ and $B_i$ the intersection $(∩ F_i) ∩ Y $ is not empty. So A is inessential on Y if there exist closed $F_i$ separating $A_i$ and $B_i$ such that $∩ F_i$ does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on which A is inessential and for every positive-dimensional Y ∩ X ╲ G there exists an infinite subfamily B ∩ A which is essential on Y.  >This theorem and its generalization provide a new approach for constructing hereditarily infinite-dimensional compacta not containing subspaces of positive dimension which are weakly infinite-dimensional or C-spaces.},
author = {Levin, Michael},
journal = {Fundamenta Mathematicae},
keywords = {inessentiality; hereditarily infinite-dimensional compacta},
language = {eng},
number = {1},
pages = {93-68},
title = {Inessentiality with respect to subspaces},
url = {http://eudml.org/doc/212077},
volume = {147},
year = {1995},
}

TY - JOUR
AU - Levin, Michael
TI - Inessentiality with respect to subspaces
JO - Fundamenta Mathematicae
PY - 1995
VL - 147
IS - 1
SP - 93
EP - 68
AB - Let X be a compactum and let $A={(A_i,B_i):i=1,2,...}$ be a countable family of pairs of disjoint subsets of X. Then A is said to be essential on Y ⊂ X if for every closed $F_i$ separating $A_i$ and $B_i$ the intersection $(∩ F_i) ∩ Y $ is not empty. So A is inessential on Y if there exist closed $F_i$ separating $A_i$ and $B_i$ such that $∩ F_i$ does not intersect Y. Properties of inessentiality are studied and applied to prove:  Theorem. For every countable family of pairs of disjoint open subsets of a compactum X there exists an open set G ∩ X on which A is inessential and for every positive-dimensional Y ∩ X ╲ G there exists an infinite subfamily B ∩ A which is essential on Y.  >This theorem and its generalization provide a new approach for constructing hereditarily infinite-dimensional compacta not containing subspaces of positive dimension which are weakly infinite-dimensional or C-spaces.
LA - eng
KW - inessentiality; hereditarily infinite-dimensional compacta
UR - http://eudml.org/doc/212077
ER -

References

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  1. [1] V. A. Chatyrko, Weakly infinite-dimensional spaces, Russian Math. Surveys 46 (3) (1991), 191-210. Zbl0743.54018
  2. [2] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, 1974. Zbl67.1092.03
  3. [3] R. Pol, Selected topics related to countable-dimensional metrizable spaces, in: General Topology and its Relations to Modern Analysis and Algebra VI, Proc. Sixth Prague Topological Symposium 1986, Academia, Prague, 421-436. 
  4. [4] R. Pol, Countable dimensional universal sets, Trans. Amer. Math. Soc. 297 (1986), 255-268. Zbl0636.54032
  5. [5] R. Pol, On light mappings without perfect fibers on compacta, preprint. Zbl0913.54015
  6. [6] L. R. Rubin, Hereditarily strongly infinite dimensional spaces, Michigan Math. J. 27 (1980), 65-73. Zbl0406.54017
  7. [7] J. J. Walsh, Infinite dimensional compacta containing no n-dimensional (n ≥ 1) subsets, Topology 18 (1979), 91-95. Zbl0408.57010

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