We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E ′ , σ(E ′ , E)) → (E ′ , β * (E ′ , E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP.We show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak * null-sequence {µn}n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP. We also define two modifications of the JNP, called the universal JNP and the JNP everywhere (briefly, the uJNP and eJNP), and thoroughly study them in the classes of locally convex spaces, Banach spaces and function spaces. We provide a characterization of the JNP in terms of operators into locally convex spaces with the uJNP or eJNP and give numerous examples clarifying relationships between the considered notions.2010 Mathematics Subject Classification. Primary 46A03; Secondary 46E10, 46E15.

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The famous Rosenthal-Lacey theorem asserts that for each infinite compact set K the Banach space C(K) admits a quotient which is either a copy of c or ℓ 2 . What is the case when the uniform topology of C(K) is replaced by the pointwise topology? Is it true that C p (X) always has an infinite-dimensional separable (or better metrizable) quotient? In this paper we prove that for a Tychonoff space X the function space C p (X) has an infinitedimensional metrizable quotient if X either contains an infinite discrete C * -embedded subspace or else X has a sequence (K n ) n∈N of compact subsets such that for every n the space K n contains two disjoint topological copies of K n+1 . Applying the latter result, we show that under ♦ there exists a zero-dimensional Efimov space K whose function space C p (K) has an infinite-dimensional metrizable quotient. These two theorems essentially improve earlier results of Kakol and Sliwa on infinite-dimensional separable quotients of C p -spaces.1991 Mathematics Subject Classification. 54C35, 54E35.

A topological space X is strongly web-compact if X admits a family Aα : α ∈ N N of relatively countably compact sets covering X and such that Aα ⊂ A β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet-Urysohn and Baire and Y is strongly web-compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web-compact is not productive. This applies to show that there are quasi-Suslin spaces X whose product X × X is not quasi-Suslin.