“Groups,” “rings,” and “lattices” are definable in the language of finitary operations and equations. “Compact Hausdorff spaces” are also equationally definable except that the requisite operations (of ultrafilter convergence) are quite infinitary. On the other hand, systems of structured sets such as “topological spaces” cannot be presented using only operations and equations. While “topological groups” is not equational when viewed as a system of sets with structure, when viewed as a system of “topological spaces with structure” the additional structure is equational; here we must say equational “over topological spaces.”
KeywordsTopological Space Category Theory Additional Structure Algebraic Theory Compact Hausdorff Space
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