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1.
A.V. Arhangel'skii 《Topology and its Applications》2007,154(3):625-634
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.
A.V. Arhangel'skii 《Topology and its Applications》2007,154(16):2950-2961
We consider the following natural questions: when a topological group G has a first countable remainder, when G has a remainder of countable tightness? This leads to some further questions on the properties of remainders of topological groups. Let G be a topological group. The following facts are established. 1. If Gω has a first countable remainder, then either G is metrizable, or G is locally compact. 2. If G has a countable network and a first countable remainder, then either G is separable and metrizable, or G is σ-compact. 3. Under (MA+¬CH) every topological group with a countable network and a first countable remainder is separable and metrizable. Some new open problems are formulated. 相似文献
3.
A. V. Arhangel'skii J. Calbrix 《Proceedings of the American Mathematical Society》1999,127(8):2497-2504
This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.
4.
A. V. Arhangel'skii R. Z. Buzyakova 《Proceedings of the American Mathematical Society》1999,127(8):2449-2458
We prove that the cardinality of every first countable linearly Lindelöf Tychonoff space does not exceed , and every strongly discretely Lindelöf Tychonoff space of countable tightness is Lindelöf.
5.
Alexander Arhangel'skii 《Topology and its Applications》2011,158(2):215-222
Some new classes of pseudoopen continuous mappings are introduced. Using these, we provide some sufficient conditions for an image of a space under a pseudoopen continuous mapping to be first-countable, or for the mapping to be biquotient. In particular, we show that if a regular pseudocompact space Y is an image of a metric space X under a pseudoopen continuous almost S-mapping, then Y is first-countable. Among our main results are Theorems 2.5, 2.11, 2.12, 2.13, 2.14. See also Example 2.15, Corollary 2.7, and Theorem 2.18. 相似文献
6.
In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2). 相似文献
7.
A.V. Arhangel'skii 《Topology and its Applications》2007,154(6):1084-1088
This article is a natural continuation of [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90]. As in [A.V. Arhangel'skii, Remainders in compactifications and generalized metrizability properties, Topology Appl. 150 (2005) 79-90], we consider the following general question: when does a Tychonoff space X have a Hausdorff compactification with a remainder belonging to a given class of spaces? A famous classical result in this direction is the well known theorem of M. Henriksen and J. Isbell [M. Henriksen, J.R. Isbell, Some properties of compactifications, Duke Math. J. 25 (1958) 83-106].It is shown that if a non-locally compact topological group G has a compactification bG such that the remainder Y=bG?G has a Gδ-diagonal, then both G and Y are separable and metrizable spaces (Theorem 5). Several corollaries are derived from this result, in particular, this one: If a compact Hausdorff space X is first countable at least at one point, and X can be represented as the union of two complementary dense subspaces Y and Z, each of which is homeomorphic to a topological group (not necessarily the same), then X is separable and metrizable (Theorem 12). It is observed that Theorem 5 does not extend to arbitrary paratopological groups. We also establish that if a topological group G has a remainder with a point-countable base, then either G is locally compact, or G is separable and metrizable. 相似文献
8.
A. V. Arhangel'skii 《Proceedings of the American Mathematical Society》2005,133(7):2165-2172
A space is said to be power-homogeneous if some power of it is homogeneous. We prove that if a Hausdorff space of point-countable type is power-homogeneous, then, for every infinite cardinal , the set of points at which has a base of cardinality not greater than , is closed in . Every power-homogeneous linearly ordered topological space also has this property. Further, if a linearly ordered space of point-countable type is power-homogeneous, then is first countable.
9.
A. V. Arhangel'skii 《Proceedings of the American Mathematical Society》2000,128(6):1881-1883
A condensation is a one-to-one onto mapping. It is established that, for each -compact metrizable space , the space of real-valued continuous functions on in the topology of pointwise convergence condenses onto a metrizable compactum. Note that not every Tychonoff space condenses onto a compactum.
10.
Alexander Arhangel'skii 《Proceedings of the American Mathematical Society》2004,132(7):2163-2170
This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) ``nice" subspaces? Our approach is based on the notion of a -space introduced by E. van Douwen and on a generalization of this notion, the notion of -space. It is proved that if a space is the union of a finite family of subparacompact subspaces, then is an -space. Under , it follows that if a separable normal -space is the union of a finite number of subparacompact subspaces, then is Lindelöf. It is also established that if a regular space is the union of a finite family of subspaces with a point-countable base, then is a -space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.