A perfectly normal nonmetrizable manifold

By a manifold we mean a Hausdorff locally Euclidean space. The purpose of this paper is to prove: THEOREM 1. If the continuum hypothesis is true, then there is a perfectly normal, Hausdorff, nonmetrizable manifold. The question of the existence of such a space was raised by R. L. Wilder [6]. There had been renewed interest in this problem because of M. Reed and P. Zenor's proof [4] that (perfectly) normal, locally compact, locally connected, Moore spaces are metrizable. However recently K. Kunen [2] used the continuum hypothesis to describe a perfectly normal, 1st countable, locally compact, hereditarily separable, non-Lindelgf topology on the line, refining the usual topology. His construction is the basic tool for our solution of Wilder's problem. Kunen combines a technique of I. Juh•tsz and A. Hajnal [1] for the construction of a similar topology on the line with a technique of A. Ostaszewski [3]. Ostaszewski used • , a combinatorial principal which holds in G6del's constructible universe and implies the continuum hypothesis, to give a perfectly normal, locally compact, countably compact, 1st countable, perfectly normal, non-Linde16f topology on co 1.Rudin first used • and Ostaszewski's techinque to construct a perfectly normal countably compact nonmetrizable manifold, but we then observed that Kunen's technique with the continuum hypothesis could also be used if one gives up countable compactness. The theorem using • is given as Corollary 2, Section III. In Section III, Theorem 2, we prove that every separable, perfectly normal manifold is hereditarily separable. Each component of a manifold is metrizable if and only if it is Lindel/Sf. Without set theoretic assumptions there are no known regular, hereditarily separable non-LindelOf spaces. The example we construct is a connected, hereditarily separable, nonmetrizable 2-manifold. A nonseparable example (with