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Given that $C(X)$ (endowed with the m-topology whose base is given by $\left\{\{f \in C(X) : |g – f|\leq u\right\}: g \in C(X) \mbox{ and $u$ is a positive unit of }C(X)\}$) is a topological ring. Can it be shown $C^*(X)$ is a topological ring with respect to relative m-topology in the following way :

Since $C(X)$ is a topological ring multiplication: $C(X) \times C(X) \to C(X)$ and subtraction: $C(X) \times C(X) \to C(X)$ are continuous on $C(X) \times C(X)$. Therefore their restrictions to $C^*(X)$ are also continuous, i.e. multiplication: $C^*(X) \times C^*(X) \to C(X)$ and subtraction: $C^*(X) \times C^*(X) \to C(X)$ are continuous. Since the images of $C^*(X) \times C^*(X)$ under multiplication and subtraction are subsets of $C^*(X)$ so multiplication: $C^*(X) \times C^*(X) \to C^*(X)$ and subtraction: $C^*(X) \times C^*(X) \to C^*(X)$ are continuous. Hence $C^*(X)$ is a topological ring with respect to the relative m-topology.

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What is $C^*(X)$? –  Martin Wanvik May 21 '12 at 16:08
    
C*(X) is the set of all bounded real-valued continuous functions on the topological space X and C(X) is the set of all real-valued continuous functions on the topological space X. –  Bilash Majumder May 22 '12 at 0:57

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