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Major Theorems in Compactness: A Unified approach

with Jerzy Dydak, Amer. Math. Monthly 99 (1992) No. 3, p. 220 - 227.
 

This paper resulted from the Research Experience for Undergraduates Program, and presents a unified approach to the following four important theorems: the Stone-Weierstrass theorem, the Tikhonov theorem, Tietze's extension theorem for normal spaces, and the (existence of) the Stone-Cech compactification. All four theorems are deduced from Uryson's lemma and a Main theorem.

In brief, the Main theorem states: For every closed subalgebra $P$ of $C\sp *(X)$ with $1\in P$, there is a compact Hausdorff space $\scr M(X)$ and a map $i\sp *\sb P(g)=g\circ i\sb P\colon C\sp *(\scr M(P))\to C\sp *(X)$ which is one-to-one with image $P$. The set $\scr M(P)$ is the set of all subalgebras $\tau$ of $P$ which are maximal with respect to the property ($*$) given $\alpha\in\tau$ the set $\alpha\sp {-1}(-\epsilon,\epsilon)$ is not empty for all $\epsilon>0$. The topology on $\scr M(P)$ has as a base $\{\scr N(\alpha)\colon\ \alpha\in P\}$, where for each $\alpha\in P$ the set $\scr N(\alpha)=\{\tau\in\scr M(P)\colon \alpha\notin\tau\}$. The authors feel that this functional approach to these four theorems, and the straightforward development of their presentation (which can be presented as a sequence of problems) provide a beneficial workout for students.

Reviewed by J. E. Vaughan