This paper, dedicated to Professor K. Morita on his sixtieth birthday, explores the properties of rings of continuous functions, specifically focusing on the ring \( C^*(X) \) of real-valued bounded continuous functions on a Tychonoff space \( X \). The author emphasizes the importance of infinite operations in these rings, such as infinite sums and joins, which are essential for characterizing topological properties like metrizability and paracompactness. The paper extends these characterizations to generalizations of metric spaces, including M-spaces and p-spaces, and discusses the relationship between \( C^*(X) \) and the uniformities of \( X \). A key theorem states that a Tychonoff space \( X \) is metrizable if and only if \( C^*(X) \) is \( \sigma \)-normally generated by a normal sequence. The paper also highlights the equivalence of certain conditions for paracompact spaces, such as being paracompact and \( M \)- or \( p \)-spaces, or being the pre-image of a metric space by a perfect mapping.This paper, dedicated to Professor K. Morita on his sixtieth birthday, explores the properties of rings of continuous functions, specifically focusing on the ring \( C^*(X) \) of real-valued bounded continuous functions on a Tychonoff space \( X \). The author emphasizes the importance of infinite operations in these rings, such as infinite sums and joins, which are essential for characterizing topological properties like metrizability and paracompactness. The paper extends these characterizations to generalizations of metric spaces, including M-spaces and p-spaces, and discusses the relationship between \( C^*(X) \) and the uniformities of \( X \). A key theorem states that a Tychonoff space \( X \) is metrizable if and only if \( C^*(X) \) is \( \sigma \)-normally generated by a normal sequence. The paper also highlights the equivalence of certain conditions for paracompact spaces, such as being paracompact and \( M \)- or \( p \)-spaces, or being the pre-image of a metric space by a perfect mapping.