On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space X, let epsilon(X) be the set of all one-point metrizable extensions of X, and when X is locally compact, let epsilon(K)(X) denote the set of all locally compact elements of epsilon(X) and lambda:epsilon(X) --> Z(beta X\X) be the order-anti-isomorphism (onto its image) defined in [M. Henriksen, L. Janos, R.G. Woods, Properties of one-point completions of a non-compact metrizable space, Comment. Math. Univ. Carolin. 46 (2005) 105-123; in short HJW]. By definition lambda(Y) = boolean AND(n>omega)cl(beta x)(U-n boolean AND X)\X, where Y = X boolean OR {p} is an element of epsilon(X) and {U-n}(n<omega) is an open base at p in Y. We characterize the elements of the image of lambda as exactly those non-empty zero-sets of beta X which miss X, and the elements of the image of epsilon(K)(X) under lambda, as those which are moreover clopen in beta X\X. This answers a question of [HJW]. We then study the relation between epsilon(X) and epsilon(K) (X) and their order structures. and introduce a subset epsilon(S)(X) of epsilon(X). We conclude with some theorems on the cardinality of the sets epsilon(X) and epsilon(K)(X), and some open questions. (C) 2006 Elsevier B.V. All rights reserved.