# The countable sup property for lattices of continuous functions

## TL;DR

This paper characterizes when certain lattices of continuous functions have the countable sup property, linking it to the topological space's chain condition, and explores how this property behaves under products and in general vector lattices.

## Contribution

It provides necessary and sufficient conditions for the countable sup property in various function lattices and examines its behavior under products, extending existing topological and lattice theory results.

## Key findings

- Countable sup property is linked to the space's countable chain condition.
- Product spaces may fail to have the property even if factors do.
- Results include new insights into vector lattices beyond continuous functions.

## Abstract

In this paper we find sufficient and necessary conditions under which vector lattice $C(X)$ and its sublattices $C_b(X)$, $C_0(X)$ and $C_c(X)$ have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space $X$. We also consider the countable sup property of $C(X\times Y)$. Even when both $C(X)$ and $C(Y)$ have the countable sup property it is possible that $C(X\times Y)$ fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when $C(\prod_{\lambda\in\Lambda} X_\lambda)$ has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest.