Domination in products

Abstract

In this note we continue to study the cardinal invariants dm and sdm introduced by the author in [7]. We prove that if K is a compact subspace of C p (Y ) for some space Y such that dm(Y ) ≤ κ then K is strongly κ-monolithic. Also we show how GCH implies that if sdm(C p (X)) ≤ ω for some hereditarily normal space X of character at most c then every infinite compact subset of X is countable. Finally we show that for every cardinal κ there is a metric space of weight at most κ that condenses onto Σ(2 κ ).

Spaces with an M-diagonal

“…The small diagonal of the compact space X implies t(X ) ≤ ω and Theorem 3.15 in [15] completes the proof.…”

“…In this section we consider the case when M is a metric space (not necessarily separable) and X is a compact space with an M-diagonal. In [15,Theorem 3.15] the author proved that if M is a metric space and X is a compact space with an M-diagonal and countable tightness, then w(X ) ≤ w(M). Corollary 3.1 Let M be a metric space and X be a compact space with an M-diagonal.…”