You have two connected topological spaces $(A,B)$. Prove that $A\times B$ is also connected.
I understand that I have to prove that there is a point in $B$ (call it $b$), that makes $A\times\{b\}$ homeomorphic to $A$ making it connected to $A\times B$. Then prove that $\{a\}\times B$ is connected in $A\times B$. But I don't really know where to being with this. If you could help that would be appreciated.
Best Answer
Let $F : A \times B \to \{0,1\}$ be a continuous functions. To show that $A\times B$ is connected for the product topology we have to show that $F$ is constant.
As you suggested (kind of) we first show that $F$ is constant on every set of the form $\{a\}\times B$. Indeed if we have $a\in A$ we get a function $f:B \to \{0,1\}$ defined by $b \mapsto F(a,b)$. This functions is continuous thus constant because $B$ is connected.
In the exact same way we can show that $F$ is constant on the sets of the form $A \times \{b\}$.
We now show that this implies that $F$ is constant on $A\times B$. Indeed fix $(a,b) \in A \times B$. Now let's consider another point $(a',b')\in A \times B$. By what we have done earlier we have $F(a,b)=F(a,b')=F(a',b')$. We are done.