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In this paper it is shown that for each positive integer n there is a locally compact Hausdorff space X having the property that dim X = n and in addition having the property that if f(X) = Y is a proper mapping, then dim Y>n. Using this result it is shown that there is a space Y having the property that min dim Y= n with a point pE Y with min dim YIP} =0. Introduction. A proper mapping is a closed continuous function f:X--Y such that f1(y) is compact for all yE Y. In this paper it is shown that for each positive integer n there is a locally compact Hausdorff space X having the property that dim X = n and in addition has the property that if f(X) = Y is a proper mapping, then dim Y> n. For n = 2 such a space was shown to exist in [51 in answer to a question posed by Isbell [4, pp. 119-120]. Using this result it is then shown that for each positive integer n there is a space X having the property that the minimum dimension of a compactification for X is n and that there is a point pCX such that X{p } has a 0-dimensional compactification. In the terminology of Isbell [4, p. 97 ], min dim X=n with min dim X{p} =0. NOTATION. Throughout the paper X and Y denote completely regular Hausdorff spaces. By dim X is meant the modified Lebesgue covering dimension of X [2, p. 243] or [4, p. 97].1 By min dim X is meant the minimum dimension of a compactification of X [4, p. 97]. The cardinals are considered to be a subset of the ordinals in the natural way. The infinite cardinals will be denoted by COO, co1, CW2, * * where wo is the first infinite ordinal, c, is the first uncountable ordinal, etc. If ae is an ordinal, then by [0, ax) will be meant the set of all ordinals less than a except that [0, 1) will denote the half-open unit interval. The set [0, a) will be thought of as a topological space with the order topology. By L(a) will be meant the set [0, a) X [0, 1) endowed with the lexicographic order and given the order topology. The space L(a) is called a long line. Received by the editors December 17, 1970. AMS 1970 subject classifications. Primary 54F45, 54D35; Secondary 54C10.

Proper mappings and the minimum dimension of a compactification of a space