| In the present work we define and study some compact-open-like topologies on the set C(X) of continuous real-valued functions defined on a topological space X and the set of reals is considered with the usual topology. The compact-open topology on C(X) has been studied by many authors and during the years many of its properties have been established. Here we introduce the notion of (compact in S)-(open in Rt ) topology on C(X), where S is a dense subset of X, and Rt is the set of reals with some topology t , not necessarily the usual metric topology. If we let S vary in some family S of dense subsets of X, and if for every S we consider the (compact in S)-(open in Rt ) topology on C(X), we get a family of topologies on C(X), for which we consider its infimum in the lattice of all topologies defined on the set C (X). Now we ask: What are the properties of the space C(X) with this topology? Since this is too general, we consider the case when X is a compact Hausdorff space with assigned filter F of dense open subsets on it, and S varies in Fd ---the set of all countable intersections of elements of F . Also, we consider only the cases when t is the usual topology or the discrete topology on the set of reals. For every S ∈ Fd we define the (compact in S)-(open in R (or Rd )) topologies and consider their infimum tF (or tF3=0 ). We study these topologies and establish some of their properties. We use them to characterize the monomorphisms in the category LSpFi of spaces with Lindelof filters, defined by R. Ball, A. Hager and A. Molitor in [2]. To the topologies we associated convergences lF and lF3=0 on C(X). These convergences have the same closure operators as tF and tF3=0 , and they are always Hausdorff l-group convergences, but only sometimes topological. These topologies are always T 1, countably tight, and homogeneous, but only sometimes Hausdorff or l-group topologies. The associations of (X, F ) to tF,t F3=0,l F , and lF3=0 are functorial. |
