Cent. Eur. J. Math. • 8(4) • 2010 • 754-762 DOI: 10.2478/s11533-010-0039-y Central European Journal of Mathematics LΣ(≤ω)-spaces and spaces of continuous functions Research Article Israel Molina Lara 1∗ , Oleg Okunev 1† 1 Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico Received 29 May 2009; accepted 13 May 2010 Abstract: We 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 c is an LΣ(≤ω)-space. MSC: 54C35, 54D20, 54C60 Keywords: Lindelöf Σ-spaces • Pointwise convergence • Compact-valued mappings • Gul’ko compact spaces © Versita Sp. z o.o. 1. Terminology, notation, and some basic facts All spaces in this article are assumed to be Tychonoff (= completely regular Hausdorff). We use terminology and notation as in [5]. The symbol κ always denotes a cardinal, finite or infinite. The symbol ω denotes the set of all natural numbers (always considered with the discrete topology), and N + = ω \{0}. c is the cardinal 2 ω . We denote C p (X,Z ) the space of all continuous functions from X to Z endowed with the topology of pointwise convergence (that is, the topology of subspace of the space Z X of all functions from X to Z equipped with the Tychonoff product topology); the space C p (X, R) is denoted as C p (X ). Given a continuous mapping g : X → Y and a space Z , the dual mapping g ∗ Z : C p (Y,Z ) → C p (X,Z ) is defined by the rule: g ∗ (f )= f ◦ g for every f ∈ C p (Y,Z ). We usually write g ∗ instead of g ∗ Z where it is clear from the context what Z is. See [2] for information on the properties of the spaces C p (X,Z ) and related mappings; we often use the facts that g ∗ Z is always continuous, and is an embedding if and only if g(X )= Y . A compact space X is an Eberlein compact space if X is homeomorphic to a subspace of C p (K ) for some compact space K ,a Gul’ko compact space if C p (X ) is a Lindelöf Σ-space, and a Corson compact space if X is homeomorphic to a subspace of a Σ-product of real lines (or, equivalently, to a subspace of C p (λ(κ)) for some κ, where λ(κ) is the one-point lindelöfication of the discrete space of cardinality κ). It is well-known that every Eberlein compact space is a Gul’ko compact space, and every Gul’ko compact space is a Corson compact space (see, e.g., [2]). For multivalued mappings we do not require that images of points all be nonempty. If p : X → Y is a multivalued mapping and A ⊂ X , then p(A) is defined as  { p(x ): x ∈ A }; a multivalued mapping p : X → Y is onto Y if p(X )= Y . ∗ E-mail: israel@molcaxitl.net † E-mail: oleg@fcfm.buap.mx 754