Graph Closures and Metric Compactifications of N

Singular Sets and Remainders

“…Let 0 = ir~\U) n f~l(V}. Then C> is a nonempty subset of /?A", and if Using 4.1 we generalize a result of Steiner and Steiner [11] which gives a sufficient condition for a compact space A" to be a remainder of A". We will call a compact space A" totally singular with respect to a compactification aX if there is a map/: X -» A so that/(U n A") is dense in K for all U which are neighborhoods of points in aA" \ X.…”

“…Perhaps the earliest (in the context of this paper) was Loeb [9]. Others who should be mentioned are Steiner and Steiner [11], [12], Magill [10], Blakley, Gerlits and Magill [1], and Choo [7]. Chandler and Tzung [6] defined the remainder induced by f to be £(/)= n{clYf(X\F)\FG%x} and proved that (whenever Y is compact) there is a compactification aX with aX \ X homeomorphic to £(/).…”