We prove that the locally convex space $C_{p}(X)$ of continuous real-valued functions on a Tychonoff $X$ equipped with topology pointwise convergence is distinguished if and only $\Delta$-space in sense \cite {Knight}. As an application this characterization theorem we obtain following results: 1) If \v{C}ech-complete (in particular, compact) such $C_p(X)$ distinguished, then scattered. 2) For ...

We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if identity map (E?, ?(E?, E)) ? ?*(E?, is not sequentially continuous. By classical theorem, every infinite-dimensional Banach has JNP. A characterization of spaces with JNP given. thoroughly study in various function spaces. Among other results we show that for Tychonoff X, Cp(X) iff there weak* null-sequence (...

This paper studies the homomorphism of rings of continuous functions ρ:C(X) → C(Y ), Y a subspace of a Tychonoff space X, induced by restriction. We ask when ρ is an epimorphism in the categorical sense. There are several appropriate categories: we look at CR, all commutative rings, and R/N, all reduced commutative rings. When X is first countable and perfectly normal (e.g., a metric space), ρ ...

Given a Tychonoff space X and classes U and V of topological groups, we say that a topological group G = G(X, U, V) is a free (U, V)-group over X if (a) X is a subspace of G, (b) GE U, and (c) every continuous f:X+ H with H EV extends uniquely to a continuous homomorphism _?: G + H. For certain classes U and V, we consider the question of the existence of free (U, V)groups. Our principal result...

One of the fundamental problem in rings continuous function is to extract those spaces for which C(X) determines X, that investigate X and Y such isomorphic with C(Y ) implies homeomorphic Y. The development started back from Tychonoff who first pointed out inevitability space this category problem. Later S. Banach M. Stone proved independently slight variance, if compact Hausdorff space, also ...

Recall that a continuous function $fcolon Xto Y$ between Tychonoff spaces is proper if and only if the Stone extension $f^{beta}colon beta Xtobeta Y$ takes remainder to remainder, in the sense that $f^{beta}[beta X-X]subseteq beta Y-Y$. We introduce the notion of ``taking remainder to remainder" to frames, and, using it, we define a frame homomorphism $hcolon Lto M$ to be $beta$-proper, $lambd...