共查询到20条相似文献,搜索用时 15 毫秒
1.
Oleg Pavlov 《Topology and its Applications》2011,158(16):2205-2209
A topological space X is called linearly Lindelöf if every increasing open cover of X has a countable subcover. It is well known that every Lindelöf space is linearly Lindelöf. The converse implication holds only in particular cases, such as X being countably paracompact or if nw(X)<ℵω.Arhangel?skii and Buzyakova proved that the cardinality of a first countable linearly Lindelöf space does not exceed ℵ02. Consequently, a first countable linearly Lindelöf space is Lindelöf if ℵω>ℵ02. They asked whether every linearly Lindelöf first countable space is Lindelöf in ZFC. This question is supported by the fact that all known linearly Lindelöf not Lindelöf spaces are of character at least ℵω. We answer this question in the negative by constructing a counterexample from MA+ℵω<ℵ02.A modification of Alster?s Michael space that is first countable is presented. 相似文献
2.
David Milovich 《Topology and its Applications》2011,158(18):2528-2534
3.
We show that the subsemigroup of the product of ω1-many circles generated by the L-space constructed by J. Moore is again an L-space. This leads to a new example of a Lindelöf topological group. The question whether all finite powers of this group are Lindelöf remains open. 相似文献
4.
Raushan Z. Buzyakova 《Topology and its Applications》2007,154(4):917-924
We describe the structure of spaces of continuous step functions over GO-spaces. We establish a relation between the Dedekind completion of a GO-space L and properties of the space of continuous functions from L to 2 with finitely many steps. We use the established relation to prove that a countably compact GO-space L has Lindelöf Cp(L) iff the Dedekind remainder of L is Lindelöf and every compact subspace of L is metrizable. Or equivalently, a countably compact GO-space L has Lindelöf Cp(L) iff every compact subspace of L is metrizable and a Gδ-set in L. Other results are obtained. 相似文献
5.
Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω12 that is an HFD with the following property
- (P)
- the projection of H onto every partial product I2 for I∈ω[ω1] is onto.
6.
We consider special subclasses of the class of Lindelöf Σ-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space X is in the class LΣ(?κ) if it admits a cover by compact subspaces of weight κ and a countable network for the cover. We restrict our attention to κ?ω. In the case κ=ω, the class includes the class of metrizably fibered spaces considered by Tkachuk, and the P-approximable spaces considered by Tka?enko. The case κ=1 corresponds to the spaces of countable network weight, but even the case κ=2 gives rise to a nontrivial class of spaces. The relation of known classes of compact spaces to these classes is considered. It is shown that not every Corson compact of weight ℵ1 is in the class LΣ(?ω), answering a question of Tkachuk. As well, we study whether certain compact spaces in LΣ(?ω) have dense metrizable subspaces, partially answering a question of Tka?enko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions. 相似文献
7.
We prove a Dichotomy Theorem: for any Hausdorff compactification bG of an arbitrary rectifiable space G, the remainder bG?G is either pseudocompact or Lindelöf. This theorem generalizes a similar theorem on topological groups obtained earlier in A.V. Arhangel'skii (2008) [6], but the proof for rectifiable spaces is considerably more involved than in the case of topological groups. It follows that if a remainder of a rectifiable space G is paracompact or Dieudonné complete, then the remainder is Lindelöf and that G is a p-space. We also present an example showing that the Dichotomy Theorem does not extend to all paratopological groups. Some other results are obtained, and some open questions are formulated. 相似文献
8.
Liang-Xue Peng 《Topology and its Applications》2012,159(1):304-307
Let μ and ν be two ordinals. If X is a subspace of μ×ν, then X is dually discrete. This gives a positive answer to a question of Alas, Junqueira and Wilson. By this conclusion and a known conclusion we show that a subspace Y of μ×ν has countable spread if and only if the space Y is hereditarily a Lindelöf D-space. 相似文献
9.
A variant of Michael's example is given to the following effect: there is a Lindelöf space M of weight ℵ1, with all Gδ-sets open, such that M×B(ℵ1) is nonnormal. This answers a question from [K. Alster, On the class of ω1-metrizable spaces whose product with every paracompact space is paracompact, Topology Appl. 153 (2006) 2508-2517]. 相似文献
10.
No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster?s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most ℵ1. It turns out that topological spaces having Alster?s property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alster?s property satisfy a seemingly stronger version of Alster?s property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property. 相似文献
11.
Mikhail Matveev 《Topology and its Applications》2010,157(7):1211-1737
It is noted that CH is equivalent to the assumption that every dense pseudocompact subspace of c2 contains a dense Lindelöf subspace. 相似文献
12.
Yankui Song 《Topology and its Applications》2011,158(9):1121-1123
In this paper, we construct an example of a T4 feebly Lindelöf space X which is not star Lindelöf under ℵ02=ℵ12, which gives a partial answer to Alas, Junqueira and Wilson (2011) [1, Question 4]. 相似文献
13.
In this article we continue the study of R-factorizability in paratopological groups. It is shown that: (1) all concepts of R-factorizability in paratopological groups coincide; (2) a Tychonoff paratopological group G is R-factorizable if and only if it is totally ω -narrow and has property ω-QU; (3) every subgroup of a T1 paratopological group G is R-factorizable provided that the topological group G? associated to G is a Lindelöf Σ-space, i.e., G is a totally Lindelöf Σ-space ; (4) if Π=∏i∈IGi is a product of T1 paratopological groups which are totally Lindelöf Σ-spaces, then each dense subgroup of Π is R-factorizable. These results answer in the affirmative several questions posed earlier by M. Sanchis and M. Tkachenko and by S. Lin and L.-H. Xie. 相似文献
14.
Assaf Rinot 《Topology and its Applications》2007,155(3):135-140
We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω1 is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC. 相似文献
15.
Debora Di Caprio 《Topology and its Applications》2006,153(14):2680-2702
We introduce and study some completeness properties for systems of open coverings of a given topological space. A Hausdorff space admitting a system of cardinality κ satisfying one of these properties is of type Gκ. Hence, we define several new variants of the ?ech number and use elementary submodels to determine further results. We introduce M-hulls and M-networks, for M elementary submodel. As an application, we give estimates for both the tightness and the Lindelöf number of a generic upper hyperspace. Two recent results of Costantini, Holá and Vitolo on the tightness of co-compact hyperspaces follow from ours as corollaries. 相似文献
16.
Yankui Song 《Topology and its Applications》2012,159(3):814-817
In this paper, we show the following statements:
- (1)
- For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.
- (2)
- Assuming ℵ02=ℵ12, there exists a centered-Lindelöf normal space X such that we(X)?ω1.
17.
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
18.
There has recently been considerable interest in productively Lindelöf spaces, i.e. spaces such that their product with every Lindelöf space is Lindelöf. See e.g. , , , , , and , and work in progress by Brendle and Raghavan. Here we make several related remarks about such spaces. Indestructible Lindelöf spaces, i.e. spaces that remain Lindelöf in every countably closed forcing extension, were introduced in [28]. Their connection with topological games and selection principles was explored in [27]. We find further connections here. 相似文献
19.
Mikhail Tkachenko 《Topology and its Applications》2009,156(7):1298-1305
We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that ?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff. 相似文献
20.
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