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## Fundamenta Mathematicae

1995 | 147 | 1 | 93-68
Tytuł artykułu

### Inessentiality with respect to subspaces

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EN
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EN
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.
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Tom
Numer
Strony
93-68
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-08-29
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autor
Bibliografia
• [1] V. A. Chatyrko, Weakly infinite-dimensional spaces, Russian Math. Surveys 46 (3) (1991), 191-210.
• [2] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, 1974.
• [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] R. Pol, Countable dimensional universal sets, Trans. Amer. Math. Soc. 297 (1986), 255-268.
• [5] R. Pol, On light mappings without perfect fibers on compacta, preprint.
• [6] L. R. Rubin, Hereditarily strongly infinite dimensional spaces, Michigan Math. J. 27 (1980), 65-73.
• [7] J. J. Walsh, Infinite dimensional compacta containing no n-dimensional (n ≥ 1) subsets, Topology 18 (1979), 91-95.
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