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Consider an embedding of $\mathbb R$ into $\mathbb R^3$ which consits of an infinite connected sum of knots, arranged roughly along the $x$-axis. There is a path in the space of proper embeddings $\mathbb R\hookrightarrow\mathbb R^3$ which goes from that knotted line to the unknotted embedding of $\mathbb R$ into $\mathbb R^3$ (where $\mathbb R$ just maps to ...


The earliest reference I could find to the fact that compact Hausdorff rings are profinite (objects in the category of topological rings) is in Johnstone's "Stone Spaces" (VI.4.11 on page 266); in the bibliographical notes on page 269 it is attributed to Kaplansky, "Topological Rings" (Amer. J. Math. 69 (1947) 153-183). Interestingly, Kaplansky himself ...


A very detailed account is given in the book "Lectures on Spectrum of $L^{2}(\Gamma \backslash G)$" by Floyd Williams. The first chapter does exactly what is required.

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