Subalgebras of  C ( X ) with the local property

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>C</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> 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.