# I-Completeness in Function Spaces

## TL;DR

This paper investigates ideal completeness in function spaces under various uniformities, establishing conditions for ideal completeness related to topological structures and convergence types.

## Contribution

It introduces new relationships between different uniformities and ideal completeness in function spaces, especially involving compactness and k-spaces.

## Key findings

- Established links between uniformity of uniform convergence on compacta and on the entire space.
- Provided sufficient conditions for ideal completeness in function spaces using k-space properties.
- Analyzed the role of I-Cauchy nets and I-convergence in the context of function space uniformities.

## Abstract

In this paper we have studied the idea of ideal completeness of function spaces Y to the power X with respect to pointwise uniformity and uniformity of uniform convergence. Further involving topological structure on X we have obtained relationships between the uniformity of uniform convergence on compacta on Y to the power X and uniformity of uniform convergence on Y to the power X in terms of I-Cauchy condition and I-convergence of a net. Also using the notion of a k-space we have given a sufficient condition for C(X,Y) to be ideal complete with respect to the uniformity of uniform convergence on compacta.