Order convergence structure on \(C(X)\). (English) Zbl 1094.46004

Anguelov, Roumen; van der Walt, Jan Harm

Quaest. Math. 28, No. 4, 425-457 (2005).

This paper brings together three concepts, the concept of order convergence, the concept of convergence space, and the concept of Hausdorff continuous functions. The primary focus is on the set \(C(X)\) of all continuous real valued functions defined on a topological space \(X\), where the order convergence is given through the natural partial order, induced in a pointwise way.
As is well known, the order convergence on a poset is generally not topological and this is indeed shown to be the case with the order convergence on \(C(X)\). It is shown that there exists a convergence structure \(\lambda_0\) on \(C(X)\) (in the sense of [R. Beattie and H.–P. Butzmann, “Convergence structures and applications to functional analysis” (Berlin: Springer) (2002; Zbl 1246.46003), which induces sequential convergence identical with the order structure. A uniform structure is induced in a natural way, however, the convergence vector space \((C(X),\lambda_0)\) is not complete. The completion is given through the larger set of Hausdorff continuous functions \(H(X)\). The obtained convergence vector space \((H(X),\lambda_0)\) may have applications, particularly to the solution of nonlinear PDEs.

Reviewer: Zoran Kadelburg (Belgrade)

MSC:

46A19	Other “topological” linear spaces (convergence spaces, ranked spaces, spaces with a metric taking values in an ordered structure more general than \(\mathbb{R}\), etc.)
46A40	Ordered topological linear spaces, vector lattices
46E05	Lattices of continuous, differentiable or analytic functions

Keywords:

order convergence; convergence space; completion

Citations:

Zbl 1246.46003