Let $X$ be a topological space. A connected component of $X$ means an equivalence class with respect to the equivalence relation on $X$ defined as:

$$x \sim y \Longleftrightarrow x, y \text { belong to some connected subspace of } X .$$

(i) Show that every connected component is a connected and closed subset of $X$.

(ii) If $X, Y$ are topological spaces and $X \times Y$ is the product space, show that every connected component of $X \times Y$ is a direct product of connected components of $X$ and $Y$.