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).

     

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.

     

Some remarks on metric spaces whose product with every Lindelöf space is Lindelöf

Let us assume that Martin's Axiom holds. We prove that if is a metrizable space whose product with every Lindelöf space is Lindelöf, then for every metric on consistent with the topology of is a countable union of totally bounded subsets.

     

Normality and paracompactness of the Fell topology

Let be a Hausdorff topological space and the hyperspace of all closed nonempty subsets of . We show that the Fell topology on is normal if and only if the space is Lindelöf and locally compact. For the Fell topology normality, paracompactness and Lindelöfness are equivalent.

     

Le degré de Lindelöf est -invariant

Two Tychonoff spaces and are said to be -equivalent if and are linearly homeomorphic. It is shown that if and are -equivalent, then the Lindelöf numbers of and are the same. The proof given is a strengthening of the one given by N.V. Velichko to show that the Lindelöf property is -invariant.

     

A first countable linearly Lindelöf not Lindelöf topological space

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.     

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.

     

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.

     

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 .

     

Productively Lindelöf and indestructibly Lindelöf spaces

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.     

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.

     

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.

     

Some of the combinatorics related to Michael's problem

We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming and that it is consistent with that there is a Michael space. The influence of Cohen reals on Michael's problem is discussed as well. Finally, we present an example of a Michael space of weight less than under the assumption that (whose product with the irrationals is necessarily linearly Lindelöf).

     

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.

     

When do connected spaces have nice connected preimages?

We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly -discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly -discrete connected topology. Another result is that a finer Tychonoff connected strictly -discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on .

     

A Lindelöf space with no Lindelöf subspace of size

A consistent example of an uncountable Lindelöf (and hence normal) space with no Lindelöf subspace of size is constructed. It remains unsolved whether extra set-theoretic assumptions are necessary for the existence of such a space. However, our space has size and is a -space, i.e., 's are open, and for such spaces extra set-theoretic assumptions turn out to be necessary.

     

On productively Lindelöf spaces

The class of spaces such that their product with every Lindelöf space is Lindelöf is not well-understood. We prove a number of new results concerning such productively Lindelöf spaces with some extra property, mainly assuming the Continuum Hypothesis.     

Some more examples of monotonically Lindelöf and not monotonically Lindelöf spaces

A space is monotonically Lindelöf (mL) if one can assign to every open cover U a countable open refinement r(U) (still covering the space) so that r(U) refines r(V) whenever U refines V. Some examples of mL and non-mL spaces are considered. In particular, it is shown that the product of a mL space and the convergent sequence need not be mL, that some L-spaces are mL, and that Cp(X) is mL only for countable X.     

Characterization for Beurling-Björck space and Schwartz space

We give an elementary proof of the equivalence of the original definition of Schwartz and our characterization for the Schwartz space . The new proof is based on the Landau inequality concerning the estimates of derivatives. Applying the same method, as an application, we give a better symmetric characterization of the Beurling-Björck space of test functions for tempered ultradistributions with respect to Fourier transform without conditions on derivatives.

     

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.