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We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if (G, ·, τ) is a regular right (left) semitopological group with dev(G) < Nov(G) such that all left (right) translations are feebly continuous, then (G, ·, τ) is a topological group. This extends several results in literature.

We consider the relationship between separately continuous functions and separately open sets, and we study the properties of the separately open topology on R2 and on Q2. We show that R2 with this topology (denoted R ⊗R) is completely and strongly Hausdorff and that Q⊗Q is normal but not a p-space. In addition, we show that each point of Q ⊗Q has an uncountable neighborhood base.

We study continuity points of functions with values in generalized metric spaces. We define the generalized oscillation, which is a useful tool in our study. Let X be a topological space and Y be a weakly developable space. Let ƒ : X →Y be a function. Then the set C (ƒ) of continuity points of ƒis a Gδ-set in X. Some results concerning continuity points of separately continuous functions as well as functions with closed graphs are also given.

Given a σ \sigma -ideal J \mathcal {J} of subsets of a space X X , mappings f : X → Y f:X \to Y are investigated, such that f | X 0 f|{X_0} is continuous for some closed X 0 ⊂ X {X_0} \subset X with X ∖ X 0 ∈ J X\backslash {X_0} \in \mathcal {J} .

Namioka proved that the following condition (*) given below holds, if X is Čech-complete and Y is a locally compact, σ-compact space.(*) Let X and Y be spaces, Z be a metric space and let f: X × Y → Z be separately continuous. Then there is a dense, Gδ set A in X such that A × Y ⊂ C(f).Following Christensen a space X is called Namioka if (*) is true for any compact space Y. In this paper we introduce and study a new class of spaces which is closely related to Namioka spaces. Namely, we say that a space Y is co-Namioka if (*) holds for any Namioka space X.

We study an analogue of symmetric quasicontinuity in the case of a function defined on the Cartesian product of three spaces. Some results of S. Kempisty, N. F. G. Martin and the second named author are generalized.

We prove some results concerning the covering, additivity, and the uniform numbers for general topological spaces. For any dense-in-itself T1 topological space X let us put cov(X) = inf{Q2: R is a family of nowhere dense subsets of X covering X}. Then cov(X) is always an infinite cardinal that does not exceed IXI, the cardinality of X. If cov(X) is an uncountable cardinal, then X is second category, and if cov(U) is uncountable for every nonempty open subset of X, then X is a Baire space. Let us put cov2(X) = inf{cov(F): F is a closed, nonempty, dense-in-itself subspace of X}. If cov2 (X) is an uncountable cardinal, then X is usually called a totally nonmeager space [1] (sometimes it is also called a totally inexhaustible space [8]). In [8], the behavior of totally nonmeager spaces under feebly continuous and feebly open mappings have been considered (see also [2]). Let us recall [4] that a mapping f of a space X onto a space Y is feebly continuous if int f-1 (V) :A 0 whenever V is nonempty and open in Y, and f is feebly open if int f(U) :A 0. whenever U is nonempty and open in X. T. Neubrunn [9] conjectured that (*) If f is a one-to-one feebly continuous and feebly open mapping of a regular space X onto a totally nonmeager space Y, then X is a Baire space. We shall show that this conjecture is true even without appealing to any additional separation axioms. For this purpose we prove the following general result. THEOREM 1. Let f be a one-to-one feebly continuous and feebly open mapping of a space X onto a space Y. If U is a nonempty open subset of X, then cov2(Y) <

A pair ( X , Y ) (X,Y) of topological spaces is said to be a Blumberg pair ("BP") if for every f : X → Y f:X \to Y , there exists a dense subset D D of X X such that f | D f|D is continuous. X X is a Blumberg space if ( X , R ) (X,R) is BP, where R R denotes the reals. Y Y is co-Blumberg if ( R , Y ) (R,Y) is BP. We survey the literature concerning the relationships between Blumberg spaces and Baire spaces and then study the relationships between co-Blumberg spaces and separability properties.