Oleg Okunev

On some classes of Lindelöf Σ-spaces

Abstract We consider special subclasses of the class of Lindelof 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 L Σ ( ⩽ 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 Tkacenko. 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 ℵ 1 is in the class L Σ ( ⩽ ω ) , answering a question of Tkachuk. As well, we study whether certain compact spaces in L Σ ( ⩽ ω ) have dense metrizable subspaces, partially answering a question of Tkacenko. Other interesting results and examples are obtained, and we conclude the paper with a number of open questions.

LΣ(≤ ω)-spaces and spaces of continuous functions

AbstractWe present a few results and problems related to spaces of continuous functions with the topology of pointwise convergence and the classes of LΣ(≤ ω)-spaces; in particular, we prove that every Gul’ko compact space of cardinality less or equal to

The one-point Lindelöfication of an uncountable discrete space can be surlindelöf

\mathfrak{c}

Pseudocomplete and weakly pseudocompact spaces

is an LΣ(≤ ω)-space.

A relation between spaces implied by their t-equivalence

We study separating function sets. We find some necessary and sufficient conditions for Cp(X) or Cp2 (X) to have a point-separating subspace that is a metric space with certain nice properties. One of the corollaries to our discussion is that for a zero-dimensional X, Cp(X) has a discrete point-separating space if and only if Cp2 (X) does.

The minitightness of products

We prove that the one-point Lindelöfication of a discrete space of cardinality ω1 is homeomorphic to a subspace of Cp(X) for some hereditarily Lindelöf space X if the axiom holds.

Calibers, \omega -continuous maps and function spaces

We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, Cp(X, M) is a continuous image of a closed subspace of Cp(X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindelöf number of Cp(X)×Cp(X) coincides with the Lindelöf number of Cp(X). We also prove that l(Cp(Xn)κ) ≤ l(Cp(X)κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.