Separation Axioms: Representation Theorems, Compactness, and Compactifications

In the historical development of general topology, the searches for appropriate compactness axioms and appropriate separation axioms are closely intertwined with each other. That such intertwining is important is proven by both the Alexandrov and Stone-Cech compactifications; that such intertwining is to be expected follows from the duality between compactness and separation—the former restricts, and the latter increases, the number of open sets; and that such intertwining is categorically necessary is proven by the categorical nature of the Stone-Cech compactification and its relationship to the Stone representation theorems. Furthermore, these compactifications and many other well-known results justify the compact Hausdorff spaces of traditional mathematics.

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