Remarks on aleph_1-metrizable not metrizable first countable spaces and CWH

Abstract

CWH, CWN stand for collectionwise Hausdorff and collectionwise normal respectively. We analyze the statement "there is a lambda-CWH not CWH first countable (Hausdorff topological) space". We prove the existence of such a space under various conditions, show its equivalence to: there is a lambda-CWN not CWN first countable space and give an equivalent set theoretic statement; the nicest version we can obtain is in \S4. The author had a flawed proof of the existence of such spaces in ZFC, for some lambda > aleph_1, in June of 1992; still we decided that there is some interest in the correct part and some additions.