On the relative cellularity of Lindelöf subspaces of topological groups

Abstract

The notions of the relative cellularity (≡ relative Souslin number) c( X, Y) and the relative τ-cellularity cel τ ( X, Y) of a subspace X of a topological space Y are introduced. Always c( X)⩾ c( X, Y)⩽cel τ ( X, Y)⩽ cel τ ( X). It is proved that cel τ ( X, G) ⩽exp τ for any τ-Lindelöf subspace X of any Hausdorff topological group G and c( X)⩽cel τ ( X)⩽ exp τ if, in addition, X is a retract of G or even a retract of some G τ-subset of G. These results are deduced form the results concerning spaces with lattices of open mappings on them.