Pseudocompact spaces $X$ and $df$-spaces $C_{c}\left ( X\right ) $

Abstract

Abstract. Let X be a completely regular Hausdorff space, and let Cc (X) be the space C (X) of continuous real-valued functions on X endowed with the compact-open topology. We find various equivalent conditions for Cc (X) to be a df -space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for Cc (X), Grothendieck's DF -spaces do not coincide with Jarchow's df -spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for Cc (X), our original motivation.

Barrelled Spaces With(out) Separable Quotients

“…Precisely when X is Warner bounded, C c (X) β is a normed space which (we show) contains 1 complemented; C c (X) β is a Banach space precisely when C c (X) is a d fspace [8]. Therefore precisely when X is Warner bounded and not compact, C c (X) is not normable (so ( * ) and ADK do not apply) and yet C c (X) β is a normed space which, by Section 3, has a separable quotient 1 , as does the completion.…”

“…Indeed, (9) ⇒ (10) is immediate. We easily argue that ¬(8) ⇒ ¬(10): if M is a closed ℵ 0 -codimensional subspace of a primitive C c (X) σ , then M ⊥ is a copy of ω in C c (X), so X is not pseudocompact [8].…”

“…Consider, for example, Morris and Wulbert's nonnormable DF-space C c [0, Ω) [14]: the strong dual is necessarily normable (and complete, in this case) with a complemented copy of 1 . In fact, examples in [7][8][9][10] show that none of the four displayed arrows can be reversed. Therefore in these examples the spaces X are nonhomeomorphic, and most are Warner bounded and not compact.…”

The Analysis of Warner Boundedness

“…admits (ε n ) n ⊂ (0, 1] such that (ε n u n ) n is bounded in E. We prove the converse for E = C c (X) σ ; for arbitrary E we do not know the answer. In [8] and [9] we showed that the bounding cardinal b (see [12]) is the largest cardinal such that the product space R κ is docile when κ < b, and gave proof for the following. Theorem 1.1.…”

“…An LCS E is docile [6][7][8][9] if every infinite-dimensional subspace of E contains an infinite-dimensional bounded set. This is weaker than the Fréchet-Urysohn property [6,Theorem 4.1], itself weaker than metrizability.…”