Paracompactness and an example due to F. B. Jones

Abstract

At the summer meeting (1955) of the American Mathematical Society, Mary E. Rudin presented an example of a separable normal nonparacompact space. It is the purpose of this note to point out that an example [3] due to F. B. Jones (1937) with an obvious definition of open sets is also such an example. Jones' paper was published before the notion of paracompactness appeared in the literature. The reader is referred to [1; 2] for definitions concerning paracompactness. EXAMPLE (JONES). Let M1 denote a subset of the open number interval I(0, 1) of cardinality K, such that each countable subset of M1 is an inner limiting set with respect to M1. Let Z1 denote the set of all points (x, y) of the number plane such that both x and y are positive rational numbers and 0 <x < 1. Furthermore, let a denote a most economical well ordered sequence of the points of M1, i.e., for p in M1, p is preceded in a by at most a countable subset of M1. Let S denote a space whose points are the points of M1 and Z1 in which open sets are defined as follows: (1) For p in Z1, p is an open set. (2) For p in M1 such that p has an immediate predecessor in a, an open set containing p is a point set D in M1+Z1 such that (a) DDp and (b) there exists an interior T of an inverted isosceles triangle with its lower vertex at p and whose base is parallel to the x-axis such that T. (M1+Z1) =D-p. (3) For p in M1 such that p has no immediate predecessor in a, let a denote a point of M1 such that a <p in a. Now, for a point x of Ml such that a <x <p in a, let T. denote the interior of an inverted isosceles triangle with its lower vertex at x and whose base is parallel to the x-axis. An open set D containing p is the set of all points y in S such that either y = x or y E T. * Zi. It is easy to see that S is a separable Hausdorff space. By a slight modification of Jones' argument, it may be shown that S is normal. In his argument where he considers K M1 to be countable, replace the set Dlk by a set Qlk such that Qlk = QM1 where Q is an open set in S such that (1) Q is countable, (2) QM1 is a closed subset of M1, and (3) Q.M1.H=QH=O.