In Galvin and Prikry's paper, they inroduce completely Ramsey sets.

Definition $5$: A set $S\subseteq 2^\mathbb{N}$ is completely Ramsey if $f^{-1}(S)$ is Ramsey for every continuous mapping $f:2^\mathbb{N}\to 2^\mathbb{N}.$

Question: What is a topology in $2^\mathbb{N}?$

As mentioned by @Patrick Stevens below, the authors consider product topology on $2^\mathbb{N}.$

I have another question:

Question: Suppose that $X$ is a finite subset of $\mathbb{N}$ and $M$ is a countable subset of $\mathbb{N}.$ Define $g:2^M\to 2^\mathbb{N}$ by $$g(A) = X\cup A.$$ How to show that $g$ is continuous?

The function $g$ is defined in the proof of Lemma $7$ of the paper above. The authors do not prove it.