On one-point metrizable extensions of locally compact metrizable spaces

For a non-compact metrizable space X, let E ( X ) be the set of all one-point metrizable extensions of X, and when X is locally compact, let E K ( X ) denote the set of all locally compact elements of E ( X ) and λ : E ( X ) → Z ( β 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 λ ( Y ) = ⋂ n < ω cl β X ( U n ∩ X ) \ X , where Y = X ∪ { p } ∈ E ( X ) and { U n } n < ω is an open base at p in Y. We characterize the elements of the image of λ as exactly those non-empty zero-sets of βX which miss X, and the elements of the image of E K ( X ) under λ, as those which are moreover clopen in β X \ X . This answers a question of [HJW]. We then study the relation between E ( X ) and E K ( X ) and their order structures, and introduce a subset E S ( X ) of E ( X ) . We conclude with some theorems on the cardinality of the sets E ( X ) and E K ( X ) , and some open questions.