Unsolved problems on algebraic aspects of C(X). (English) Zbl 0587.54028

Henriksen, Melvin E.

Rings of continuous functions, Pap. Spec. Sess. Annu. Meet. Am. Math. Soc., Cincinnati/Ohio 1982, Lect. Notes Pure Appl. Math. 95, 195-202 (1985).

[For the entire collection see Zbl 0559.00006.]
The paper is a brief sketch of some old and new problems concerning algebraic and topological aspects of the ring C(X) of all continuous real-valued functions on a Tikhonov space X. We list here a few of them. Problem 1. When is a \(\Phi\)-algebra A isomorphic to C(X) for some (realcompact) X? Problem 3. Is the space of minimal prime ideals P(X) (with the hull-kernel topology) always basically disconnected? Problem 4. Characterize local connectedness of a space X in terms of the ring C(X). The paper contains a comprehensive list of references.

Reviewer: M.G.Tkachenko

MSC:

54C40

Algebraic properties of function spaces in general topology

Keywords:

Freudenthal compactification; rimcompact space; realcompact spaces; weak P-point; \(\Phi \) -algebra; minimal prime ideals; hull-kernel topology; local connectedness

Citations:

Zbl 0559.00006