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P. Szewczak 《Topology and its Applications》2011,158(2):177-182
In [3] R. Telgársky (1975) asked: does the first player have a winning strategy in the game G(F,X×X) if the first player has a winning strategy in the game G(F,X)? I give a positive answer to this question and prove that this result is also true for spaces where the first player has a winning strategy in game G(K,X) where K=1, F, C, for σC if X is P-space and for DC if X is collectionwise-normal space. The last result is related to the Telgársky's (1983) conjecture discussed in [1]. These results are not true for infinite product of spaces. 相似文献
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It is shown that each Σ-product of paracompact p-spaces has the weak
-property. 相似文献
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Peter Nyikos Leszek Piatkiewicz 《Proceedings of the American Mathematical Society》1996,124(1):303-314
In 1975 E. K. van Douwen showed that if is a family of Hausdorff spaces such that all finite subproducts are paracompact, then for each element of the box product the -product is paracompact. He asked whether this result remains true if one considers uncountable families of spaces. In this paper we prove in particular the following result: Let be an infinite cardinal number, and let be a family of compact Hausdorff spaces. Let be a fixed point. Given a family of open subsets of which covers , there exists an open locally finite in refinement of which covers . We also prove a slightly weaker version of this theorem for Hausdorff spaces with ``all finite subproducts are paracompact" property. As a corollary we get an affirmative answer to van Douwen's question.
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Guram Bezhanishvili Nick Bezhanishvili Joel Lucero-Bryan Jan van Mill 《Annals of Pure and Applied Logic》2019,170(5):558-577
For a topological space X, let be the modal logic of X where □ is interpreted as interior (and hence ◇ as closure) in X. It was shown in [3] that the modal logics S4, S4.1, S4.2, S4.1.2, S4.Grz, (), and their intersections arise as for some Stone space X. We give an example of a scattered Stone space whose logic is not such an intersection. This gives an affirmative answer to [3, Question 6.2]. On the other hand, we show that a scattered Stone space that is in addition hereditarily paracompact does not give rise to a new logic; namely we show that the logic of such a space is either S4.Grz or for some . In fact, we prove this result for any scattered locally compact open hereditarily collectionwise normal and open hereditarily strongly zero-dimensional space. 相似文献
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There is a paracompact Q-set space in ZFC 总被引:1,自引:0,他引:1
Zoltan T. Balogh 《Proceedings of the American Mathematical Society》1998,126(6):1827-1833
We construct a paracompact space such that every subset of is an -set, yet is not -discrete. We will construct our space not to have a -diagonal, which answers questions of A.V. Arhangel'skii and D. Shakhmatov on cleavable spaces.
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Jerzy Dydak 《Topology and its Applications》2004,140(2-3):227-243
We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper:
Theorem. Suppose X is a paracompact space. There is a CW complex K such that
- (a) K is an absolute extensor of X up to homotopy,
- (b) If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy.
Theorem. Let X be a paracompact space. Suppose a space Y is the union of a family {Ys}sS of its subspaces with the following properties:
- (a) Each Ys is an absolute extensor of X,
- (b) For any two elements s and t of S there is uS such that YsYtYu.
If f :A→Y is a map from a closed subset A to Y such that A=sSIntA(f−1(Ys)), then f extends over X.
That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own. 相似文献
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Yukinobu Yajima 《Proceedings of the American Mathematical Society》2003,131(4):1297-1302
The separation property in our title is that, for two spaces and , any two disjoint closed copies of in are separated by open sets in . It is proved that a Tychonoff space is paracompact if and only if this separation property holds for the space and every Tychonoff space which is a perfect image of (where denotes the Stone-Cech compactification of ). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.
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K. Alster 《Topology and its Applications》2009,156(7):1345-1347
We prove that if X is a paracompact space which has a neighborhood assignment x→Hx such that for each y∈X the closure of the set is compact then the products T×X, for every paracompact space T, and Xω are paracompact. The first result answers a problem of H. Junnila. 相似文献
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