Let $A\subset$ $\mathbb{R}^n$ and $B\subset$ $\mathbb{R}^m$ $(A,B \neq \emptyset )$.
Show and prove if those propositions are true or false:
a) $A ×B $ is compact $\iff$ $A$ and $B$ are compact.
b) $A × B$ are connected $\iff$ $A$ and $B$ are connected.
c) $A × B$ are path-connected $\iff$ $A$ and $B$ are path-connected.
I have tried but I dont know how to do it correctly. It is the first time I write here so I'm sorry if have mistakes with Latex.
a): Suppose that $A\times B$ is compact. Consider $p_A:A\times B\rightarrow A$, it is continue. So if $A\times B$ is compact, so is $p_A(A\times B)=A$ since the image of a compact space by a continuous map is compact.
Suppose that $A,B$ compact, apply Tichonof.
b) Suppose that $A\times B$ is compact Let $f:A\rightarrow \{0,1\}$ be a continuous map, $f\circ p_A$ is continue, since $A\times B$ is connected, it is constant, so $f$ is constant since $p_A$ is surjective.
Suppose $A,B$ connected. Let $f:A\times B\rightarrow \{0,1\}$ be a continuous map, for every $a\in A$ let $f_a:B\rightarrow \{0,1\}$ defined by $f_a(b)=f(a,b)$, since $A$ is connected, $f_a$ is constant.
For $b\in B$, defined $g_b:A\rightarrow B$ by $g_b(a)=f(a,b)$, $g_b$ is constant since $A$ is connected. Let $a_0\in A,b_0\in B$. For every $a\in A,b\in B$, we have $f(a_0,b_0)=f_{a_0}(b_0)=f_{a_0}(b)=g_b(a_0)=g_b(a)=f(a,b)$. This implies that $f$ is constant and $A\times B$ connected.
c) Suppose that $A\times B$ is path connected, for every $a,a'\in A$, there exists a path $f:[0,1]\rightarrow A\times B$ such that $f(0)=(a,b)$ and $f(1)=(a',b)$. The map $p_A\circ f$ is a path between $a$ and $a'$.
Suppose that $A$ and $B$ are path connected, let $a,a'\in A$ and $b,b'$ in $B$, there exists a path $c:[0,1]\rightarrow A$ such that $c(0)=a, c(1)=a'$ and a path $d:[0,1]\rightarrow B$ such that $d(0)=b, d(1)=b'$. The map $e:[0,1]\rightarrow A\times B$ defined by $e(t)=(c(t),d(t))$ is a continuous path between $(a,b)$ and $(a',b')$.