Small Diagonals
by
Gary Gruenhage
Auburn University

According to M. Husek, a space X has a small diagonal if for every uncountable subset Y of X²\\Delta, there is an open set U containing \Delta such that Y\U is uncountable. Clearly a space with a G_\delta-diagonal has a small diagonal; the question is, does small diagonal imply G_\delta-diagonal (for certain classes of spaces)? This question for compact T₂-spaces is a well-known and still not completely solved problem of Husek, though Juhasz has shown that CH implies a positive answer. Here we show: (1) the answer for countably compact spaces depends on your set theory; (2) there are consistent examples of regular hereditarily Lindelöf, and consistent with CH examples of regular Lindelöf, spaces with a small diagonal but no G-diagonal; (3) there is in ZFC a locally compact space with a small diagonal but no G_\delta-diagonal.

Date received: June 7, 1999