Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$
That's the problem 11.1 from commutative algebra course
As answered here
we can see the bijection between $\operatorname{Spec}(A_1\times A_2)$ and $\operatorname{Spec}(A_1)\sqcup\operatorname{Spec}(A_2)$ as follows: the prime ideals of $A_1\times A_2$ are of the form $\mathfrak{p}\times A_2$ where $\mathfrak p$ is a prime ideal of $A_1$, or $A_1\times\mathfrak q$ where $\mathfrak q$ is a prime ideal of $A_2$.
That means that we should find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x])$
As discussed here: Let $A$ be a commutative ring with unit, $X = \operatorname{Spec}(A) $ with the Zariski topology. The irreducible components are $\lbrace V(p) : p\subset A \ \text{minimal prime ideal} \rbrace$ where $V(P) =\lbrace q \ \text{prime ideal } \mid p\subset q\rbrace$.
An ideal is called prime if the complement is a multiplicative set.
I know that prime ideals of $\mathbb{C}[x]$ are the principal ideals generated by $\lbrace (x-a) | a \in \mathbb{C}\rbrace$ and $0$. $0$ is contained in all the other ideals.
The prime ideals $\mathfrak{p}$ of $\mathbb{C}[x]$ except $0$ are all minimal, so the irreducible components of $\operatorname{Spec}\mathbb{C}[x]$ are all $\lbrace V(p) \rbrace$.
Is the final answer that the connected component are the same as irreducible components and are in the bijection with $I_{x-a} \times \mathbb{C}[y]$ and $\mathbb{C}[x] \times I_{y-a}$ ?
Could you explain how this topology can be seen "in a picture"?