I am trying to understand the Compactness Argument in a Graph Theory Problem using Probabilistic Methods. $V$ is infinite set.
For each finite subset $X \subset V$, let $C_X \subset [2]^V$ be the collection of all coloring using 2 colors in the Vertex set $X$ where no edge contained in $X$ is monochromatic. It is said that $C_X$ for $ |X| < \infty$ is closed in the Product Topology
but I don’t understand why its closed?
Thank you in advance.
Write $$ S_v^k = \{f \in [2]^V \,:\, f(v)=k\} $$ for the set of colorings where a particular vertex is a particular color. Note that every $S_v^k$ is closed: $S_v^k$ is precisely the preimage $\pi_v^{-1}(\{k\})$ under the $v$th projection map $\pi_v\colon [2]^V\to [2]$. The singleton $\{k\}$ is closed in the discrete set $[2]$, and the projection from a product down to one of the factors is continuous, so our preimage of a closed set is closed.
In general, $$ C_X = \bigcap_{\{a,b\} \in \binom{X}{2}\cap E} \bigg((S_a^0 \cap S_b^1) \cup (S_a^1 \cap S_b^0)\bigg), $$ since the requirement is that every edge $(a,b)$ in the induced subgraph is either (red, blue) colored or (blue, red) colored. If $X$ is finite then the above intersection is finite. Finite unions and intersections of closed sets are closed, so $C_X$ is closed.