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1.
J.A. Brown 《Topology and its Applications》2008,155(4):190-200
Assume CH. Let I be any index set, and let Xi, for i∈I, be a completely regular ccc topological space of weight ω2. If X=∏i∈IXi is ccc and non-pseudocompact, then X has remote points. 相似文献
2.
Let H denote the halfline [0,∞). A point p?βH?H is called a near point if p is in the closure of some countable discrete closed subspace of H. In addition, a point p?βH?H is called a large point if p is not in the closure of a closed subset of H of finite Lebesgue measure. We will show that for every autohomeomorphism ? of βH?H and for each near point p we have that ?(p) is not large. In addition, we establish, under CH, the existence of a point x?βH?H such that for each autohomeomorphism ? of βH?H the point ?(x) is neither large nor near. 相似文献
3.
Alan Dow 《Topology and its Applications》1983,15(3):239-246
It is shown that ω × Yω does not have remote points if Y is a compact space with cellularity larger than ω1. It is also shown that it is consistent that ω × Yω does not have remote points if Y is compact with uncountable cellularity. As an application we construct a compact space with weight ω2 · c which can be covered by nowhere dense P-sets and a compact space with weight c for which it is independent that it can be covered by nowhere dense P-sets. 相似文献
4.
In this work we expand upon the theory of open ultrafilters in the setting of regular spaces. In [E. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981) 1-45], van Douwen showed that if X is a non-feebly compact Tychonoff space with a countable π-base, then βX has a remote point. We develop a related result for the class of regular spaces which shows that in a non-feebly compact regular space X with a countable π-base, there exists a free open ultrafilter on X that is also a regular filter.Of central importance is a result of Mooney [D.D. Mooney, H-bounded sets, Topology Proc. 18 (1993) 195-207] that characterizes open ultrafilters as open filters that are saturated and disjoint-prime. Smirnov [J.M. Smirnov, Some relations on the theory of dimensions, Mat. Sb. 29 (1951) 157-172] showed that maximal completely regular filters are disjoint prime, from which it was concluded that βX is a perfect extension for a Tychonoff space X. We extend this result, and other results of Skljarenko [E.G. Skljarenko, Some questions in the theory of bicompactifications, Amer. Math. Soc. Transl. Ser. 2 58 (1966) 216-266], by showing that a maximal regular filter on any Hausdorff space is disjoint prime.Open ultrafilters are integral to the study of maximal points and lower topologies in the partial order of Hausdorff topologies on a fixed set. We show that a maximal point in a Hausdorff space cannot have a neighborhood base of feebly compact neighborhoods. One corollary is that no locally countably compact Hausdorff topology is a lower topology, which was shown previously under the additional assumption of countable tightness by Alas and Wilson [O. Alas, R. Wilson, Which topologies can have immediate successors in the lattice of T1-topologies? Appl. Gen. Topol. 5 (2004) 231-242]. Another is that a maximal point in a feebly compact space is not a regular point. This generalizes results of both Carlson [N. Carlson, Lower upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007) 619-624] and Costantini [C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008) 187-225]. 相似文献
5.
Désirée Basile 《Topology and its Applications》2008,155(4):180-189
In [A.V. Arhangel'ski?, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90], Arhangel'ski? introduced the notion of Ohio completeness and proved it to be a useful concept in his study of remainders of compactifications and generalized metrizability properties. We will investigate the behavior of Ohio completeness with respect to closed subspaces and products. We will prove among other things that if an uncountable product is Ohio complete, then all but countably many factors are compact. As a consequence, Rκ is not Ohio complete, for every uncountable cardinal number κ. 相似文献
6.
We show that, under CH, (i) every ccc, non-pseudocompact space of weight at most ω2 has remote points, and (ii) if Xi, for i ∈ I, is a completely regular ccc topological space of weight ω1 and if X := Πi ∈ I Xi is ccc and non-pseudocompact, then X has remote points.
Received July 21, 2006; accepted in final form October 16, 2008. 相似文献
7.
8.
K.L. Kozlov 《Topology and its Applications》2010,157(4):698-707
The approach to the problem of the distribution of the functors of the Stone-?ech compactification, the Hewitt realcompactification or the Dieudonné completion with the operation of taking products is discussed using uniform structures on products. In particular, the role of different rectangular conditions is shown. Relative analogues of this question and new examples of (strongly) rectangular products are presented. Characterizations of bounded rectangular subsets of the product are given. 相似文献
9.
Sophia Zafiridou 《Topology and its Applications》2008,156(1):142-149
We investigate the problem of existence of universal elements in some families of dendrites with a countable closure of the set of end points. In particular, we prove that for each integer κ?3 and for each ordinal α?1 there exists a universal element in the family of all dendrites X such that ord(X)?κ and the α-derivative of the set clXE(X) contains at most one point. 相似文献
10.
M.N. Mukherjee 《Topology and its Applications》2007,154(18):3167-3172
In the present paper, a kind of extension, termed ideal extension of a given topological space is considered via the concept of ideals. A general method of construction of such an extension of a T0—space is worked out and it is finally shown that under certain condition imposed on the ideals involved, the said extension space turns out to be the compactification of a given space. 相似文献
11.
Takamitsu Yamauchi 《Topology and its Applications》2008,155(8):916-922
It is shown that if X is a countably paracompact collectionwise normal space, Y is a Banach space and φ:X→Y2 is a lower semicontinuous mapping such that φ(x) is Y or a compact convex subset with Cardφ(x)>1 for each x∈X, then φ admits a continuous selection f:X→Y such that f(x) is not an extreme point of φ(x) for each x∈X. This is an affirmative answer to the problem posed by V. Gutev, H. Ohta and K. Yamazaki [V. Gutev, H. Ohta and K. Yamazaki, Selections and sandwich-like properties via semi-continuous Banach-valued functions, J. Math. Soc. Japan 55 (2003) 499-521]. 相似文献
12.
《Quaestiones Mathematicae》2013,36(2):171-175
Abstract If every infinite closed subset of the Wallman compactification, WX, of a space X must contain at least one element of X, then for any space Y intermediate between X and WX the Wallman compactification WY is homeomorphic to WX. This extends a property which characterizes normality inducing spaces. In the case where X is not normal, however, this is not a characterization, since there are nonnormal spaces for which all intermediate spaces are Wallman equivalent, but have infinite closed subsets contained in WX/X. 相似文献
13.
Mateusz Krukowski 《Quaestiones Mathematicae》2018,41(3):349-357
In the paper, we recall the Wallman compactification of a Tychonoff space T (denoted by Wall(T)) and the contribution made by Gillman and Jerison. Motivated by the Gelfand-Naimark theorem, we investigate the homeomorphism between Cb(T), the space of continuous and bounded functions on T , and C(Wall(T)), the space of continuous functions on the Wallman compactification of T. Along the way, we attempt to justify the advantages of the Wallman compactification over other manifestations of the Stone-?ech compactification. The main result of the paper is a new form of the Arzelà-Ascoli theorem, which introduces the concept of equicontinuity along ω-ultrafilters. 相似文献
14.
《Quaestiones Mathematicae》2013,36(1-2):109-116
Abstract We show that a B-conjunctive frame L, where B is a normal base for L gives rise to a strong inclusion on L and therefore a compactification of L. The resulting compact regular frame corresponds to the quotient frame obtained by Johnstone in his construction of the Wallman compactification for frames. It is also shown that, in the presence of pseudocompactness the Wallman compactification and the Wallman realcompactification coincide. 相似文献
15.
Lajos Soukup 《Topology and its Applications》2008,155(4):347-353
Nagata conjectured that every M-space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. Although this conjecture was refuted by Burke and van Douwen, and A. Kato, independently, but we can show that there is a c.c.c. poset P of size ω2 such that in VP Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space X∈V is an M-space in VP then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in VP). By a result of Morita, it is enough to show that every first countable regular space from the ground model has a first countable countably compact extension in VP. As a corollary, we also obtain that every first countable regular space from the ground model has a maximal first countable extension in model VP. 相似文献
16.
Eva Murtinová 《Topology and its Applications》2006,153(18):3402-3408
Assuming a measurable cardinal exists, we construct a pair of discretely generated spaces whose product fails to be weakly discretely generated. Under the Continuum Hypothesis, a similar result is obtained for a pair of countable Fréchet spaces as well as for two compact discretely generated spaces whose product is not discretely generated. A somewhat weaker example is presented assuming Martin's Axiom for countable posets. Further, the class of strongly discretely generated compacta is shown to preserve discrete generability in products. 相似文献
17.
Tetsuya Ishiu 《Topology and its Applications》2008,155(11):1256-1263
We proved that ?+ implies the existence of a non-D-space whose all closed subspace F satisfies e(F)=L(F). The existence of such a space under MA+¬CH or PFA is also discussed. 相似文献
18.
M.R. Koushesh 《Topology and its Applications》2011,158(16):2191-2197
For a locally pseudocompact space X let
ζX=X∪clβX(βX\υX). 相似文献
19.
Nathan Carlson 《Topology and its Applications》2007,154(3):619-624
In the partial order of Hausdorff topologies on a fixed infinite set there may exist topologies τ?σ in which there is no Hausdorff topology μ satisfying σ?μ?τ. τ and σ are lower and upper topologies in this partial order, respectively. Alas and Wilson showed that a compact Hausdorff space cannot contain a maximal point and therefore its topology is not lower. We generalize this result by showing that a maximal point in an H-closed space is not a regular point. Furthermore, we construct in ZFC an example of a countably compact, countably tight lower topology, answering a question of Alas and Wilson. Finally, we characterize topologies that are upper in this partial order as simple extension topologies. 相似文献
20.
M.R. Koushesh 《Topology and its Applications》2011,158(3):509-532
A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y′ of X let Y?Y′ if there is a continuous function of Y′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.
Conjecture.
For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic. 相似文献