Normality and dense subspaces

In the first section of this paper, using certain powerful results in -theory, we show that there exists a nice linear topological space of weight such that no dense subspace of is normal. In the second and third sections a natural generalization of normality, called dense normality, is considered. In particular, it is shown in section 2 that the space is not normal on some countable dense subspace of it, while it is normal on some other dense subspace. An example of a Tychonoff space , which is not densely normal on a dense separable metrizable subspace, is constructed. In section 3, a link between dense normality and relative countable compactness is established. In section 4 the result of section 1 is extended to densely normal spaces.

     

Lindelöf property and absolute embeddings

It is proved that a Tychonoff space is Lindelöf if and only if whenever a Tychonoff space contains two disjoint closed copies and of , then these copies can be separated in by open sets. We also show that a Tychonoff space is weakly -embedded (relatively normal) in every larger Tychonoff space if and only if is either almost compact or Lindelöf (normal almost compact or Lindelöf).

     

Characterizations of paracompactness and Lindelöfness by the separation property

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.

     

Fell-拓扑的伪紧性（英文）

设X是拓扑空间，CL（X）表示X的所有非空闭子集的族，本文得到了下述结果：在CL（X）上的Fell-拓扑是伪肾的当且仅当X是feebly-紧或者非局部紧或者非σ－紧，由此得到了对于伪紧性不是闭遗传的两类新的拓扑空间。     

Countable network weight and multiplication of Borel sets

A space Borel multiplies with a space if each Borel set of is a member of the -algebra in generated by Borel rectangles. We show that a regular space Borel multiplies with every regular space if and only if has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for to Borel multiply with every metric space.

     

Fell topology on the space of functions with closed graph

LetX andY beT ₁ topological spaces andG(X, Y) the space of all functions with closed graph. Conditions under which the Fell topology and the weak Fell topology coincide onG(X,Y) are given. Relations between the convergence in the Fell topologyτF, Kuratowski and continuous convergence are studied too. Characterizations of a topological spaceX by separation axioms of (G(X, R), τF) and topological properties of (G(X, R), τF) are investigated.     

On linearly Lindelöf and strongly discretely Lindelöf spaces

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.

     

Pseudocompact Whyburn spaces need not be Fréchet

We prove in ZFC that there exists a Tychonoff pseudocompact scattered AP-space of uncountable tightness. We give some sufficient and necessary conditions for a -space to be AP as well as a characterization of AP-property in linearly ordered topological spaces.

     

No submaximal topology on a countable set is -complementary

Two -topologies and given on the same set , are called transversal if their union generates the discrete topology on . The topologies and are -complementary if they are transversal and their intersection is the cofinite topology on . We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no -complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).

     

$L$-Fuzzy拓扑空间中的可数紧性度和Lindel\"{o}f性质度

In this paper,the concept of countable compactness degree and the concept of Lindelf property degree are defined in L-fuzzy topological spaces by means of implication operator →.Many properties of them are discussed.     

Affine and homeomorphic embeddings into

It is shown that

(1)

a locally compact convex subset of a topological vector space that admits a sequence of continuous affine functionals separating points of affinely embeds into a Hilbert space;

(2)

an infinite-dimensional locally compact convex subset of a metric linear space has a central point;

(3)

every -compact locally convex metric linear space topologically embeds onto a pre-Hilbert space.

     

Lindelöf powers and products of function spaces

We give criteria for finite and countable powers of a space similar to the Michael line being Lindelöf. As applications, we give examples related to Lindelöf property in products of spaces of Michael line type and in products of spaces of continuous functions on separable -compact spaces.

     

On spread and condensations

A space has a property strictly if every finite power of has . A condensation is a one-to-one continuous mapping onto. For Tychonoff spaces, the following results are established. If the strict spread of is countable, then can be condensed onto a strictly hereditarily separable space. If , then can be condensed onto a strictly hereditarily separable space, and therefore, every compact subspace of is strictly hereditarily separable. Under , if is a topological group such that , then is strictly hereditarily Lindelöf and strictly hereditarily separable.

     

On fuzzy compact-open topology

The concept of fuzzy compact-open topology is introduced and some characterizations of this topology are discussed.     

Failure of the Denjoy theorem for quasiregular maps in dimension

In 1929 L. V. Ahlfors proved the Denjoy conjecture which states that the order of an entire holomorphic function of the plane must be at least if the map has at least finite asymptotic values. In this paper, we prove that the Denjoy theorem has no counterpart in the classical form for quasiregular maps in dimensions . We construct a quasiregular map of with a bounded order but with infinitely many asymptotic limits. Our method also gives a new construction for a counterexample of Lindelöf's theorem for quasiregular maps of .

     

Fell topology on the hyperspace of a non-Hausdorff space

Fell topology is very widely used today, even in metric spaces; but J. Fell introduced it in a non-Hausdorff context in the connection with the theory of C ^*-algebras. In spite of this, it has been studied only on the hyperspace of a Hausdorff space, except for the first results due to Fell himself. The present paper aims to fill this gap, in particular extending some results of H. Poppe and of G. Beer to the general case. Project 10251002 supported by National Natural Science Foundation of China.     

On condensations of -spaces onto compacta

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.

     

A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

     

Computational estimation of the order of

The paper describes a search for increasingly large extrema (ILE) of in the range . For , the complete set of ILE (57 of them) was determined. In total, 162 ILE were found, and they suggest that . There are several regular patterns in the location of ILE, and arguments for these regularities are presented. The paper concludes with a discussion of prospects for further computational progress.