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We first determine a necessary and sufficient condition for a function      f ∈  C ∗  ( X ) f \in {C^\ast }(X)      , which extends to a compactification of      X X      , to extend to a smaller compactification. We apply this result to show that when      | β  X ∖  X | ≦  ℵ  0 |\beta X\backslash X| \leqq {\aleph _0}       there is an      f ∈  C ∗  ( X ) f \in {C^\ast }(X)       which extends to no compactification other than      β  X \beta X      . Two examples show that when      ℵ  0 &gt; | β  X ∖  X | ≦  c {\aleph _0} &gt; |\beta X\backslash X| \leqq c       no such definite result may be obtained.

The compactifications to which an element of $C\sp{\ast} (X)$\ extends