# Inessentiality with respect to subspaces

Fundamenta Mathematicae (1995)

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

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topLevin, 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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- [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.
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- [5] R. Pol, On light mappings without perfect fibers on compacta, preprint. Zbl0913.54015
- [6] L. R. Rubin, Hereditarily strongly infinite dimensional spaces, Michigan Math. J. 27 (1980), 65-73. Zbl0406.54017
- [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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