Many Eberlein–Grothendieck spaces have no non-trivial convergent sequences

Abstract

We establish that a monolithic compact space X is not scattered if and only if Open image in new window has a dense subset without non-trivial convergent sequences. Besides, for any cardinal \(\kappa \geqslant \mathfrak {c}\), the space \(\mathbb {R}^\kappa \) has a dense subspace without non-trivial convergent sequences. If X is an uncountable \(\sigma \)-compact space of countable weight, then any dense set Open image in new window has a dense subspace without non-trivial convergent sequences. We also prove that for any countably compact sequential space X, if Open image in new window has a dense k-subspace, then X is scattered.