No abstract

ABSTRACT. This paper studies two topologies on [6] to be the proper setting to study sequences of functions which converge uniformly on compact subsets. One of the distinguishing features of this topology is that whenever x is locally compact the compact-open topology on C(X) is the coarsest topology making the evaluation map e:X C(X)--. continuous (where e(x,])= f(x)).The compact-open topology and the topology of uniform convergence are equal if and only if x is compact. Because compactness is such a strong condition, there is a considerable gap between these two topologies. This gap was especially felt in [8] while studying the completeness of a normed linear space of continuous linear functionals on C(X) with the compact-open topology.Because of this, a new class of topologies was introduced in [7] on C*(X) to bridge the gap, where c*(x) is the set of bounded functions in C(x). This also generalized the a-compact-open topology

In the definition of a set-open topology on C(X), the set of all real-valued continuous functions on a Tychonoff space X, we use a certain family of subsets of X and open subsets of R. But instead of using this traditional way to define topologies on C(X), in this paper, we adopt a different approach to define two interesting topologies on C(X). We call them the open-point and the bi-point-open topologies and study the separation and countability properties of these topologies.

ABSTRACT. This is a study of the completeness properties of the space C ps (X) of continuous real-valued functions on a Tychonoff space X, where the function space has the pseudocompact-open topology. The properties range from complete metrizability to the Baire space property.

A ring [Formula: see text] is defined to be feebly clean, if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text], [Formula: see text] are orthogonal idempotents. Feebly clean rings generalize clean rings and are also a proper generalization of weakly clean rings. The family of all semiclean rings properly contains the family of all feebly clean rings. Further properties of feebly clean rings are studied, some of them analogous to those for clean rings. The feebly clean property is investigated for some rings of complex-valued continuous functions. Throughout, all rings are commutative with identity.