Topology + Foundations
32nd Summer Conference on Topology and Its Applications
We show it consistent for spaces X and Y to be both HS and HL even though their product X ×Y contains an S-space. Recall that an S-space is a T3 space that is HS but not HL.
More generally, consider spaces that contain neither an S-space nor an L-space. We say a space is ESLC iff each of its subspaces is either both HS and HL or neither HS nor HL. The "C" in "ESLC" refers to HC; a space is HC iff each of its subspaces has the ccc (countable chain condition) (iff the space has no uncountable discrete subspaces). Classes of ESLC spaces include metric spaces (because every metric space is either second countable or has an uncountable discrete subspace), subspaces of the Sorgenfrey line and (suitably defined) generalized butterfly spaces; for these classes, countable products are still ESLC.
Copyright © 2017, the Authors
Hart, Joan and Kunen, Kenneth, "Spaces with No S or L Subspaces" (2017). Summer Conference on Topology and Its Applications. 31.