Let $x,y\in \mathbb{Z}^\omega$ and let $x,y\in\mathbb{Z}^\omega$ form an edge if there is $i\in\omega$ such that $|x_i - y_i|=1$ and $ x_k = y_k$ for all $k\in \omega\setminus\{i\}$.
$K_\omega$, the complete graph on $\omega$ points, is a minor of $\mathbb{Z}^\omega$ (see this post).
Question. Is it possible that there is a cardinal $\lambda>\omega$ such that $K_\lambda$ is a minor of $\mathbb{Z}^\omega$?
If I understand the definition correctly, the answer is no.
If $K_\lambda$ is a minor of $G$, then there is a 1-1 map from $K_\lambda$ into the family of (nonempty) connected subsets of $G$, such that any two images are in the same connected component of $G$, but disjoint. So there must be a component of size $\ge \lambda$.
If you take $G$ to be $\mathbb Z^\omega$, then every point in $G$ has only countably many neighbors, so all connected components are countable.
