How many connected components could this set have?

Question

Suppose $U \subset \mathbb R^2$ is an open subset. Let $(x,y)$ denote the coordinates. For any $(x_0, y_0) \in U$, if we fix $x_0$, the set $U_{x_0} = \{y \in \mathbb R: (x_0, y) \in U\}$ has two connected components; if we fix $x_0$, the set $U_{y_0} = \{x \in \mathbb R: (x, y_0) \in U\}$ has two connected components. How many connected components could $U$ have? Is it $4$?