Cell-like maps onto non-compact spaces of finite cohomological dimension

A metrizable space X is the cell-like image of a metrizable space Z of dimension ⩽ n iff the cohomological dimension X ⩽ n. If X is topologically complete and/or separable, then we may choose Z to be so. If X is a metrizable space with coh. dim. ⩽ n, then X can be embedded in a topologically complete metrizable space with coh. dim. ⩽ n.     

On metrizable non-Archimedean LF-spaces

It is known that no non-Archimedean LB-space (and no strict non-Archimedean LF-space) is metrizable. We show that there exist many metrizable (or even normable) non-Archimedean LF-spaces. We prove that every non-normable polar non-Archimedean Fréchet space (and every non-Archimedean Banach space with an infinite basis (x_α)) contains a dense subspace which is an LF-space.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

Metrization of Fpp-spaces

A Hausdorff space each subspace of which is a paracompact p-space is an F_pp-space. A space X is a closed hereditary Baire space if each closed subspace of X is a Baire space. Using a delicate theorem of Z. Balogh it is shown that a first-countable F_pp-space that is a closed hereditary Baire space is metrizable.     

On fuzzy metrizability

In this paper we answer two questions raised by M. A. Erceg [J. Math. Anal. Appl. 69 (1979), 205–230]: precisely we show that the fuzzy unit interval is never T₀ except in the standard case and that a fuzzy pseudo-metrizable T₀ space is metrizable in the sense of Erceg (ibid.); hence the two definitions of metrizable space given in B. Hutton and I. Reilly [Fuzzy Sets and Systems3 (1980), 93–104] and Erceg (ibid.) are equivalent.     

Cosmic spaces which are not μ-spaces among function spaces with the topology of pointwise convergence

A space is called a μ-space if it can be embedded in a countable product of paracompact F_σ-metrizable spaces. The following are shown:(1) For a Tychonoff space X, if C_p(X,R) is a μ-space, then X is a countable union of compact metrizable subspaces.(2) For a zero-dimensional space X, C_p(X,2) is a μ-space if and only if X is a countable union of compact metrizable subspaces.In particular, let P be the space of irrational numbers. Then C_p(P,2) is a cosmic space (i.e., a space with a countable network) which is not a μ-space.     

A counterexample to Wood's conjecture

We prove that if the one-point compactification of a locally compact, noncompact Hausdorff space L is the topological space called pseudoarc, then C₀(L,C) is almost transitive. We also obtain two necessary conditions on a metrizable locally compact Hausdorff space L for C₀(L) being almost transitive.     

Spaces of measurable functions

For a metrizable space X and a finite measure space (Ω, $\mathfrak{M}$ , µ), the space M _µ(X) of all equivalence classes (under the relation of equality almost everywhere mod µ) of $\mathfrak{M}$ -measurable functions from Ω to X, whose images are separable, equipped with the topology of convergence in measure, and some of its subspaces are studied. In particular, it is shown that M _µ(X) is homeomorphic to a Hilbert space provided µ is (nonzero) nonatomic and X is completely metrizable and has more than one point.     

Variational principles in non-metrizable spaces

We study generic variational principles in optimization when the underlying topological space X is not necessarily metrizable. It turns out that, to ensure the validity of such a principle, instead of having a complete metric which generates the topology in the space X (which is the case of most variational principles), it is enough that we dispose of a complete metric on X which is stronger than the topology in X and fragments the space X.     

On the metrizability of cone metric spaces

We have shown in this paper that a (complete) cone metric space (X,E,P,d) is indeed (completely) metrizable for a suitable metric D. Moreover, given any finite number of contractions f₁,…,fn on the cone metric space (X,E,P,d), D can be defined in such a way that these functions become also contractions on (X,D).     

The Baire category theorem in products of linear spaces and topological groups

A space is a Baire space if the intersection of countably many dense open sets is dense. We show that if X is a non-separable completely metrizable linear space (pathconnected abelian topological group) then X contains two linear subspaces (subgroups) E and F such that both E and F are Baire but E×F is not. If X is a completely metrizable linear space of weight ℵ₁ then X is the direct sum E⊕F of two linear subspaces E and F such that both E and F are Baire but E×F is not.     

Compact coverings for Baire locally convex spaces

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space Cp(X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff Cp(X) is Baire (i.e. of the second Baire category) and is covered by a family of compact sets such that Kα⊂Kβ if α?β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.     

On sequentially compact T2 compactifications

It is known that a compact space can fail to be sequentially compact. In this paper we consider the following problem: when does a space admit a sequentially compact T₂ compactification? In the first section we develop a method to produce such compactifications, and we apply it in the second section to study the question using coverings.Moreover, we obtain solutions for locally compact T₂ spaces, and for metrizable spaces.     

Metrizability of spaces of holomorphic functions

In this paper we prove that if U is an open subset of a metrizable locally convex space E of infinite dimension, the space H(U) of all holomorphic functions on U, endowed with the Nachbin-Coeuré topology τδ, is not metrizable. Our result can be applied to get that, for all usual topologies, H(U) is metrizable if and only if E has finite dimension.     

Left K-Completeness in Quasi-Metric Spaces

Regular left K-sequentially complete quasi-metric spaces are characterized. We deduce that these spaces are complete Aronszajn and that every metrizable space admitting a left K-sequentially complete quasi-metric is completely metrizable. We also characterize quasi-metric spaces having a quasi-metric left K-sequential completion in terms of certain bases of countable order.     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

On the small diagonals

In this paper we prove that each Hausdorff compact (resp. T₃ countably compact) space with a small (resp. regularly small) diagonal is metrizable under CH+FA. Several ‘nice’ spaces, which have a small diagonal, but no G_δ-diagonal, are given under MA+¬CH. Some applications to the metrization problem are also obtained.     

There is no upper bound of small transfinite compactness degree in metrizable spaces

In [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993], Aarts and Nishiura investigated several types of dimensions modulo a class P of spaces. These dimension functions have natural transfinite extensions. The small transfinite compactness degree trcmp is such transfinite dimension function extending the small compactness degree cmp. We shall prove that there is no upper bound for trcmp in the class of metrizable spaces, i.e. for each ordinal number α there exists a metrizable space Xα such that trcmpXα=α. We also give a characterization of the dimension dim of a separable (compact) metrizable space in terms of the function cmp of the product of this space with a nowhere locally compact zero-dimensional factor.     

Abelian extensions of topological vector groups

The possibility of endowing an Abelian topological group G with the structure of a topological vector space when a subgroup F of G and the quotient group $\text{G}\text{F}$ are topological vector groups is investigated. It is shown that, if F is a real Fréchet group and $\text{G}\text{F}$ a complete metrizable real vector group, then G is a complete metrizable real vector group. This result is of particular interest if $\text{G}\text{F}$ is finite dimensional or if F is one dimensional and $\text{G}\text{F}$ a separable Hilbert group.     

Inverse systems and I-favorable spaces

We show that a compact space is I-favorable if, and only if it can be represented as the limit of a σ-complete inverse system of compact metrizable spaces with skeletal bonding maps. We also show that any completely regular I-favorable space can be embedded as a dense subset of the limit of a σ-complete inverse system of separable metrizable spaces with skeletal bonding maps.