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1.
We introduce a new cardinal invariant, core of a space, defined for any locally compact Hausdorff space X and denoted by cor(X). Locally compact spaces of countable core generalize locally compact σ-compact spaces in a way that is slightly exotic, but still quite natural. We show in Section 1 that under a broad range of conditions locally compact spaces of countable core must be σ-compact. In particular, normal locally compact spaces of countable core and realcompact locally compact spaces of countable core are σ-compact. Perfect mappings preserve the class of spaces of countable core in both directions (Section 2). The Alexandroff compactification aX is weakly first countable at the Alexandroff point a if and only if cor(X)=ω (Section 3). Two examples of non-σ-compact locally compact spaces of countable core are discussed in Section 3. We also extend the well-known theorem of Alexandroff and Urysohn on the cardinality of perfectly normal compacta to compacta satisfying a weak version of perfect normality. Several open problems are formulated.  相似文献   

2.
In this article we investigate which compact spaces remain compact under countably closed forcing. We prove that, assuming the Continuum Hypothesis, the natural generalizations to ω1ω1-sequences of the selection principle and topological game versions of the Rothberger property are not equivalent, even for compact spaces. We also show that Tall and Usuba?s “11-Borel Conjecture” is equiconsistent with the existence of an inaccessible cardinal.  相似文献   

3.
Let S be the class of all spaces, each of which is homeomorphic to a stationary subset of a regular uncountable cardinal (depending on the space). In this paper, we prove the following result: The product X×C of a monotonically normal space X and a compact space C is normal if and only if S×C is normal for each closed subspace S in X belonging to S. As a corollary, we obtain the following result: If the product of a monotonically normal space and a compact space is orthocompact, then it is normal.  相似文献   

4.
Within the framework of Zermelo-Fraenkel set theory ZF, we investigate the set-theoretical strength of the following statements:
(1)
For every family(Ai)iIof sets there exists a family(Ti)iIsuch that for everyiI(Ai,Ti)is a compactT2space.
(2)
For every family(Ai)iIof sets there exists a family(Ti)iIsuch that for everyiI(Ai,Ti)is a compact, scattered, T2space.
(3)
For every set X, every compactR1topology (itsT0-reflection isT2) on X can be enlarged to a compactT2topology.
We show:
(a)
(1) implies every infinite set can be split into two infinite sets.
(b)
(2) iff AC.
(c)
(3) and “there exists a free ultrafilter” iff AC.
We also show that if the topology of certain compact T1 spaces can be enlarged to a compact T2 topology then (1) holds true. But in general, compact T1 topologies do not extend to compact T2 ones.  相似文献   

5.
We construct a non-paracompact Hausdorff space for which ?ech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic, and our space admits non-numerable principal bundles.  相似文献   

6.
We show that every KC space (X,τ), such that τ is minimal among the KC topologies on X, must be compact (not necessarily T2). This solves a long-standing question, first raised by R. Larson in 1973.  相似文献   

7.
We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.  相似文献   

8.
We study the question whether a topological space X with a property P can be embedded in a countably compact space X? with the same property P.  相似文献   

9.
It is well known that every pair of disjoint closed subsets F0,F1 of a normal T1-space X admits a star-finite open cover U of X such that, for every UU, either or holds. We define a T1-space X to be strongly base-normal if there is a base B for X with |B|=w(X) satisfying that every pair of disjoint closed subsets F0,F1 of X admits a star-finite cover B of X by members of B such that, for every BB, either or holds. We prove that there is a base-normal space which is not strongly base-normal. Moreover, we show that Rudin's Dowker space is strongly base-(collectionwise)normal. Strong zero-dimensionality on base-normal spaces are also studied.  相似文献   

10.
F.B. Jones has proved that for many different topological properties P if there exists a non-normal space with property P then there exists a non-completely regular space Y with property P. In this paper we study the topological structure of the space Y and we characterize the topological spaces with a similar structure to that possessed by Y.  相似文献   

11.
We introduce zero-dimensional proximities and show that the poset 〈Z(X),?〉 of inequivalent zero-dimensional compactifications of a zero-dimensional Hausdorff space X is isomorphic to the poset 〈Π(X),?〉 of zero-dimensional proximities on X that induce the topology on X. This solves a problem posed by Leo Esakia. We also show that 〈Π(X),?〉 is isomorphic to the poset 〈B(X),⊆〉 of Boolean bases of X, and derive Dwinger's theorem that 〈Z(X),?〉 is isomorphic to 〈B(X),⊆〉 as a corollary. As another corollary, we obtain that for a regular extremally disconnected space X, the Stone-?ech compactification of X is a unique up to equivalence extremally disconnected compactification of X.  相似文献   

12.
A neighbourhood assignment in a space X is a family of open subsets of X such that xOx for any xX. A set YX is a kernel ofO if . We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371-377].  相似文献   

13.
We show in a direct way that a space is D if it is a finite union of subparacompact scattered spaces. This result cannot be extended to countable unions, since it is known that there is a regular space which is a countable union of paracompact scattered spaces and which is not D. Nevertheless, we show that every space which is the union of countably many regular Lindelöf C-scattered spaces has the D-property. Also, we prove that a space is D if it is a locally finite union of regular Lindelöf C-scattered spaces.  相似文献   

14.
We present instances of the following phenomenon: if a product of topological spaces satisfies some given compactness property then the factors satisfy a stronger compactness property, except possibly for a small number of factors.The first known result of this kind, a consequence of a theorem by A.H. Stone, asserts that if a product is regular and Lindelöf then all but at most countably many factors are compact. We generalize this result to various forms of final compactness, and extend it to two-cardinal compactness. In addition, our results need no separation axiom.  相似文献   

15.
16.
If X and Y are locally compact GO spaces then X×Y is dually discrete. If μ and ν are two ordinals and X is a normal subspace of μ×ν then X is dually discrete.  相似文献   

17.
We study realcompactness in the classes of submaximal and maximal spaces. It is shown that a normal submaximal space of cardinality less than the first measurable is realcompact. ZFC examples of submaximal not realcompact and maximal not realcompact spaces are constructed. These examples answer questions posed in [O.T. Alas, M. Sanchis, M.G. Tka?enko, V.V. Tkachuk, R.G. Wilson, Irresolvable and submaximal spaces: homogeneity versus σ-discreteness and new ZFC examples, Topology Appl. 107 (3) (2000) 259-273] and generalize some results from [D.P. Baturov, On perfectly normal dense subspaces of products, Topology Appl. 154 (2) (2007) 374-383].  相似文献   

18.
19.
In this note we prove that every Eberlein compact linearly ordered space is metrizable. (By an Eberlein compact space we mean a topological space which can be embedded as a compact subset of a Banach space with the weak topology.)  相似文献   

20.
We prove a preservation theorem for the class of Valdivia compact spaces, which involves inverse sequences of retractions of a certain kind. Consequently, a compact space of weight?1 is Valdivia compact iff it is the limit of an inverse sequence of metric compacta whose bonding maps are retractions. As a corollary, we show that the class of Valdivia compacta of weight?1 is preserved both under retractions and under open 0-dimensional images. Finally, we characterize the class of all Valdivia compacta in the language of category theory, which implies that this class is preserved under all continuous weight preserving functors.  相似文献   

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