Examples separating total countable compactness variants and ω-boundedness

Construct topological spaces witnessing the separations: (a) Construct a space that is totally countably compact for finite sets—i.e., for every sequence of finite subsets (F_n) of the space there exists an infinite A ⊆ ω such that the union ⋃_{n∈A} F_n is compact—but is not ω-bounded (there exists a countable subset whose closure is not compact). (b) Construct a space that is totally countably compact—i.e., for every sequence (C_n) of subsets of the space there exists an infinite A ⊆ ω such that {C_n : n ∈ A} is compact—but is not totally countably compact for finite sets.

Background

Section 2.1 introduces strengthened notions of countable compactness tailored for later Tukey-order arguments: totally countably compact, totally countably compact for finite sets, and ω-bounded. These properties form a strict implication chain (ω-bounded ⇒ totally countably compact for finite sets ⇒ totally countably compact ⇒ countably compact).

The authors connect these compactness notions to when the pairs (X, K(X)) or (F(X), K(X)) Tukey quotient to countable directed sets, playing a key role in eliminating extraneous ×ω factors in subsequent Tukey computations. The problem asks for explicit examples separating these closely related notions.