Vol. 27, No. 1, 1968

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Algebraic properties of certain rings of continuous functions

Li Pi Su

Vol. 27 (1968), No. 1, 175–191

Let X and Y be any subsets of En, and (X,d1) and (Y ,d2) be any metric spaces. Let Cm(X), 0 m , denote the ring of m-differentiable functions on X, and Lc(X) be the ring of the functions which are Lipschitzian on each compact subset of X, and L(X) be the ring of the bounded Lipschitzian functions on X. The relations between algebraic properties of Cm(X), (resp. Lc(X) or L(X) and the topological properties of X (resp. X) are studied. It is proved that if X and Y , (resp. (X,d1) and (Y ,d2)) are m-realcompact, (resp. Lc-real-compact or compact) then Cm(X)Cm(Y ) (resp. Lc(X)Lc(Y ) or L(X)L(Y ) if and only if X and Y are Cm-diffeomorphic (resp. (X,d1) and (Y ,d2) are Lc or L-homeomorphic).

Mathematical Subject Classification
Primary: 46.55
Received: 21 February 1967
Published: 1 October 1968
Li Pi Su