I'm not well-versed in topology and encountered this as part of a larger argument about coloring the real numbers:
"An arbitrary Cartesian product of compact topological spaces is compact (Tychonoff's Theorem), and in particular, the space $M$ of all mappings $f: \mathbb{R} \rightarrow [k]$ is compact." (Here, $k$ is a positive integer and $[k] = \{1, \ldots, k\}$.)
The topology they define on this space is "that of the Cartesian power $[k]^{\mathbb{R}}$"; I am not familiar with this topology.
I don't see the connection between Tychonoff's Theorem and the statement. In other words, in what sense is $M$ a Cartesian product of compact topological spaces? This confuses me because the elements of $M$ are mappings, and it is not clear to me how a mapping itself could be an element of a Cartesian product.