A theorem on rings of continuous functions.
Staknis, Victor Richard
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Given two compact Hausdorff topological spaces X and Y and the corresponding normed rings Rx and Ry of continuous real-valued functions on X and Y, respectively, we consider relationships between the function space Y^X of continuous mappings of X into Y and a certain subset of the function space RxRy of continuous homomorphisms of Ry into Rx. The so concerned subset is the set of all elements of RxRy which map Ry onto analytic subrings of Rx. An analytic subring is a closed subring which contains the constant functions. This subset is denoted by (RxRy)*. [TRUNCATED] As a consequence of the main result we obtain: a necessary and sufficient condition for two mappings f and g of X into Y to be homotopic is that f* and g* be homotopic. Where f and g correspond to f* and g*, respectively, under F. From this we obtain as a corollary: a necessary and sufficient condition that a compact Hausdorff space X be contractible is that the identity isomorphism of Rx onto Rx be homotopic to an homomorphism of Rx onto the constant functions.
Thesis (Ph.D.)--Boston University