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I-uniform continuity and I-uniform boundedness of a function

Ghosal S.

Demonstratio Mathematica

2012

Vol. 45, nr 4

887-894

The concepts of I-convergence and I-Cauchy condition are a generalization of statistical convergence and statistical Cauchy conditions and are dependent on the notion of the ideal I of subsets of the set N of positive integers. In this paper, we shall introduce two new notions of I-uniform continuity and I-uniform boundedness of a function with values in R or in a metric space and then study their basic properties.

Correspondences between ideals and z-filters for rings of continuous functions between C* and C

Panman P., Sack J., Watson S.

Commentationes Mathematicae

2012

Vol. 52, [Z] 1

11-20

Let X be a completely regular topological space. Let A(X) be a ring of continuous functions between C(X) and C{X), that is, C*{X) ⊆ A(X) ⊆ C(X). In [9], a correspondence Z_A between ideals of A(X) and z-filters on X is defined. Here we show that Z A extends the well-known correspondence for C* (X) to all rings A(X). We define a new correspondence 3 A and show that it extends the well-known correspondence for C(X) to all rings A{X). We give a formula that relates the two correspondences. We use properties of Z A. and 3 A to characterize C* (X) and C(X) among all rings A(X). We show that 3A defines a one-one correspondence between maximal ideals in A(X) and the z-ultrafilters in X.