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A space X is said to have property (USC) (resp. (LSC)) if whenever { f n : n ∈ ω } is a sequence of upper (resp. lower) semicontinuous functions from X into the closed unit interval [ 0 , 1 ] converging pointwise to the constant function 0 with the value 0, there is a sequence { g n : n ∈ ω } of continuous functions from X into [ 0 , 1 ] such that f n ⩽ g n ( n ∈ ω ) and { g n : n ∈ ω } converges pointwise to 0. In this paper, we study spaces having these properties and related ones. In particular, we show that (a) for a subset X of the real line, X has property (USC) if and only if it is a σ-set; (b) if X is a space of non-measurable cardinal and has property (LSC), then it is discrete. Our research comes of Scheepers' conjecture on properties S 1 ( Γ , Γ ) and wQN.

Let l , m , n be integers such that 0 ⩽ l ⩽ n and 0 m ⩽ n . We show that there is a first countable, separable, ω 1 -compact, countably paracompact and normal space X l , m , n with ind X l , m , n = l , dim X l , m , n = m and Ind X l , m , n = n and point out some interesting questions concerning the gap between the dimensions of paracompact spaces.

We present an example of a σ -product that is not countably paracompact but all of whose finite subproducts are countably paracompact. This example also shows that countable paracompactness of a σ -product may depend on the choice of base point. We also show that normal non-trivial σ -products are countably paracompact, improving a result of Chiba. Finally we give a new proof that σ -products of ordinals at base point 0 are κ -normal and strongly zero-dimensional.

We construct, in ZFC, a hereditarily collectionwise normal, hereditarily metaLindelof, hereditarily realcompact Dowker space. This answers a question of R. Hodel (also asked by S. Watson and D. Burke) and another question of M.E. Rudin.

We will see that: (1) In ZFC, for each subspace X⫅ω 2 1 , the following are equivalent; (a) X is normal, (b) X is countably paracompact and strongly collectionwise Hausdorff, (c) X is expandable. (2) Under a variety of different set-theoretic assumptions (including V = L and PMEA) all countably paracompact subspaces of ω 2 1 are normal. (3) All subspaces of ω 2 1 are collectionwise Hausdorff.

Let X and Y be Tychonoff spaces and C(X, Y) be the space of all continuous functions from X to Y. The coincidence of the fine topology with other function space topologies on C(X, Y) is discussed. Also cardinal invariants of the fine topology on C(X, R ) , where R is the space of reals, are studied. To answer some questions of Di Maio and Naimpally (1992) other function space topologies are investigated, namely, Krikorian topology, open-cover topology, graph topology, topology of uniform convergence, proximal graph topology.

We introduce and study several subclasses of the class of σ-spaces. The smallest of the classes considered, that of LF-netted spaces, contains all F σ -discrete spaces and all stratifiable F σ -metrizable spaces. The main result of the paper establishes the equivalence of normality and countable paracompactness of the product of an LF-netted space with a countably paracompact and normal space.

First, as an analogue of Dowker's theorem for countable paracompactness, we prove a characterization of countable metacompactness in terms of subnormality of products. Second, as an analogue of Tamano's theorem for paracompactness, we give a characterization of Lindelofness in terms of normality of products.

Let A and B be subspaces of the initial segment of an uncountable ordinal number κ with the order topology. We prove: 1. (i) For the product A × B, the following properties (1)-(3) are equivalent: (1) shrinking property; (2) collectionwise normality; and (3) normality. 2. (ii)For the product A × B, the following properties (4)-(7) are equivalent: (4) strong D- property; (5) expandability; (6) countable paracompactness; and (7) weak D(ω)-property. 3. (iii) If κ = ω1, then for A × B, the properties (1)-(7) above are equivalent, and A × B has one of them iff A or B is not stationary, or A ∩ B is stationary. 4. (iv) If κ is regular and A and B are stationary, then A × Bis κ-compact iff A ∩ B is stationary.

We correct the proof of Theorem 8 in “Normality and countable paracompactness of hyperspaces of ordinals” [Topology Appl. 154 (2007) 358–362].

We construct two examples the first of which is a Lindelof, separable and strongly zero- dimensional space the increment of which in any compactification is collectionwise normal, countably paracompact and infinite dimensional. The second example is a Lindelof, separable, non-semicompact space that has a compactification with discrete increment.

It is proven that under V = L, every locally compact, countably paracompact space is collectionwise normal with respect to compact sets. From this it can be shown that under V = L, submetacompact implies paracompact in the class of locally compact, countably paracompact spaces.

The continuum hypothesis implies that there are (normal) countably paracompact spaces X and Y such that the product X × Y is normal and not countably paracompact.

Let A and B be subspaces of an ordinal. It is proved that the product A × B is countably paracompact if and only if it is rectangular. Before this main result, we discuss several covering properties of products with one ordinal factor. In particular, for every paracompact space X , it is proved that the product X × A is paracompact if so is A .

For an ordinal α , 2 α denotes the collection of all nonempty closed sets of α with the Vietoris topology and K ( α ) denotes the collection of all nonempty compact sets of α with the subspace topology of 2 α . It is well known that 2 α is normal iff cf α = 1 . In this paper, we will prove that for every nonzero-ordinal α : (1) 2 α is countably paracompact iff cf α ≠ ω . (2) K ( α ) is countably paracompact. (3) K ( α ) is normal iff, if cf α is uncountable, then cf α = α . In (3), we use elementary submodel techniques.