Abstract A subalgebra A of C (X) has the local property if the functions in C (X) that agree locally with functions in A are also in A. We study the relationship between this property and the properties of being closed under uniform convergence, closed under inversion, and closed under composition with functions in C (R) C (R). We prove that if αX is a compactification of X, and C α (X) is the subalgebra of C * (X) consisting of the functions in C (X) that have a continuous extension to αX, then the functions in C (X) that are locally in C α (X) are those functions that can be continuously extended to some open set of αX containing X. We also prove that the uniform closure of any subalgebra and sublattice of C (X) with the local property is a subalgebra and sublattice of C (X) that is closed under inversion.
Domínguez et al. (2026) studied this question.