In this paper, we show that the existence of certain first-countable compact-like extensions is equivalent to the equality between corresponding cardinal characteristics of the continuum. For instance, $\mathfrak b=\mathfrak s=\mathfrak c$ if and only if every regular first-countable space of weight $< \mathfrak c$ can be densely embedded into a regular first-countable countably compact space.

Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X, there exists a subspace A⊆X with property P such that X=St(A,𝒰), where St(A,𝒰)=⋃ {U∈𝒰:U∩A≠0̸}. In this paper we construct an example of a pseudocompact Tychonoff space that is not star Lindelöf, which gives a negative answer to Alas et al. [‘Countability and star covering properties’, Topology Appl.158 (2011), 620–626, Question 3].

In answer to Jarchow’s 1981 text, we recently characterized when $C_{\textrm{c}}(X)$ is a $df$-space, finding along the way attractive analytic characterizations of when the Tychonov space $X$ is pseudocompact. Analogues now reveal how exquisitely Warner boundedness lies between these two notions. To illustrate, $X$ is pseudocompact, $X$ is Warner bounded or $C_{\textrm{c}}(X)$ is a $df$-space if and only if for each sequence $(\mu_{n})_{n}\subset C_{\textrm{c}}(X)'$ there exists a sequence $(\varepsilon_{n})_{n}\subset(0,1]$ such that $(\varepsilon_{n}\mu_{n})_{n}$ is weakly bounded, is strongly bounded or is equicontinuous, respectively. Our characterizations and proofs add to and simplify Warner’s.

AMS 2000 Mathematics subject classification: Primary 46A08; 46A30; 54C35

This presentation concerns the relation of chain conditions on a space X, with the weights of compact sets in Cp(X), generalizing up to the class of dσ-bounded spaces, or stable spaces. In the last case, stronger results are obtained for Corson compact subsets of CP(X).