I'm sure this is a simple question but I just can't wrap my head around the inclusion map.
In an example it says that $i_1: A \rightarrow A \times B$ and $i_2: B \rightarrow A\times B$ are the inclusions of the first and second factors respectively.
So I'm assuming an element $a \in A$ gets sent to $(a,*)$ and an element $b \in B$ gets sent to $(*,b)$ but I'm not sure what $*$ will be in these 2 cases. Is it just the basepoints $b_0$ and $a_0$ respectively or to every element of $B$ and $A$ respectively to make up the whole set $A \times B$? Or am I just completely misunderstanding this concept?
When we talk about an inclusion map, we start with topological spaces $S \subset T$ and then construct the map to be the canonical one: $\phi:S \hookrightarrow T$ defined such that $x \mapsto x$ for all $x \in S$. The thing of fundamental importance here is that $S$ actually is a subset of $T$; it is this that makes possible the adjective "canonical".
Sometimes, authors will be less-than-formal with their terminology, and you'll hear people talk about, for example, the "inclusion" $\Phi: \mathbb{R} \hookrightarrow \mathbb{R}^2$ defined such that $x \mapsto (x, 0)$. But notice the difference: $\mathbb{R} \not\subset \mathbb{R}^2$. Zero was chosen somewhat arbitrarily, mostly because people will look at the plane and think of the set $\{(x, 0) \ | \ x \in \mathbb{R} \}$ as "the real line", which isn't perfectly correct of course, but the ideas conveyed through an abuse of terminology are nevertheless well-preserved (usually), with the advantage of brevity.
If we have an "inclusion", e.g. $\mathbb{R} \hookrightarrow \mathbb{R}^2$, or more generally, $A \hookrightarrow A \times B$, it is more accurate to call this a topological embedding--that is, a map $\phi: A \hookrightarrow A \times B$ where $\phi$ is a homeomorphism onto its image. This is always possible in such scenarios, so indeed the target space contains a topologically identical copy of the space being "injected" by $\phi$ (which is why it can be tempting to think of $A$ as a subset of $A \times B$). There are many ways to define this map. With the $\mathbb{R}^2$ example, we could've just as easily defined $\Phi$ to be such that $x \mapsto (x, y)$ for any $y \in \mathbb{R}$.