A Stone-Čech compactification for limit spaces

O. Wyler [Notices Amer. Math. Soc. 15 (1968), 169. Abstract #653-306.] has given a Stone-Čech compactification for limit spaces. However, his is not necessarily an embedding. Here, it is shown that any Hausdorff limit space $(X,\tau )$ can be embedded as a dense subspace of a compact, Hausdorff, limit space $({X_1},{\tau _1})$ with the following property: any continuous function from $(X,\tau )$ into a compact, Hausdorff, regular limit space can be uniquely extended to a continuous function on $({X_1},{\tau _1})$.