A large number of papers have been published that are devoted to showing that certain algebraic objects obtained by defining operations on the set of all continuous real-valued functions on a suitably restricted topological space determine the space. We mention but a few of them in this article.
© 1956 American Mathematical Society
Henriksen, Melvin. "On the equivalence of the ring, lattice, and semigroup of continuous functions." Proceedings of the American Mathematical Society 7.6 (December 1956): 959-960. DOI: 10.1090/S0002-9939-1956-0082490-3