Let X be completely regular and Hausdorff. The family !%(X) of all those spaces which are remainders of X in compactifications has attracted a considerable amount of interest. In this paper we determine, in a certain sense, the family 3t(X) for a fairly large number of spaces X. 1. Statement of main theorem and a corollary. We state at the outset that all topological spaces under discussion here are assumed to be completely regular and Hausdorff. Moreover, a countable set in this paper is one

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... ich is in one-to-one correspondence with the natural numbers. In particular, finite sets will not be regarded as being countable. A remainder of a space X is any space of the form aX-X where aX is a compactification of X. For any space X, we use the symbol ¿%(X) to denote the family of all remainders of X. In other words, ¿%(X) consists precisely of those spaces with which X can be compactified. The main result of this paper shows that, for many spaces X, £%(X) consists of all continuous images of the free union of a finite number of copies of a familiar space. We recall that the free union of a family of spaces {Xn : n e A} is the set (J {Xn : ne A} topologized by defining G<= (J {Xn : n e A} to be open if and only if G n Xn is open for each « e A. We use the symbol F+ to denote the space of nonnegative real numbers and, as is customary, the Stone-Cech compactification of a space X will be denoted by ßX. Now we are in a position to state the main Theorem. Suppose X is noncompact, locally compact, locally connected and metric and â?(X) contains no countable spaces. Then there exists a positive integer N such that âl(X) consists precisely of all continuous images of the free union of N copies of ßR+-R + . We denote the cardinality of any set X by card X. In [4, p. 91], it is shown that card ßR = card ßN where F denotes the space of real numbers and N denotes the