This is an announcement for the paper "Inverse limits and function algebras" by Joan E. Hart and Kenneth Kunen. Abstract: Assuming Jensen's principle diamond, there is a compact Hausdorff space X which is hereditarily Lindelof, hereditarily separable, and connected, such that no closed subspace of X is both perfect and totally disconnected. The Proper Forcing Axiom implies that there is no such space. The diamond example also fails to satisfy the CSWP (the complex version of the Stone-Weierstrass Theorem). This space cannot contain the two earlier examples of failure of the CSWP, which were totally disconnected -- specifically, the Cantor set (W. Rudin) and beta N (Hoffman and Singer). Archive classification: General Topology; Functional Analysis Mathematics Subject Classification: 54D05; 46J10 Remarks: 16 pages The source file(s), invlim.tex: 45668 bytes, is(are) stored in gzipped form as 0504228.gz with size 15kb. The corresponding postcript file has gzipped size 80kb. Submitted from: kunen@math.wisc.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.GN/0504228 or http://arXiv.org/abs/math.GN/0504228 or by email in unzipped form by transmitting an empty message with subject line uget 0504228 or in gzipped form by using subject line get 0504228 to: math@arXiv.org.