Question

Prove that a countable product of connected space is connected

Answer 1

Let $\phi:A×B\rightarrow \{0,1\}$ be a continuous function. It suffices to show that $\phi$ is constant.

Now if we fix $a\in A$ then we get a function $f:B \rightarrow \{0,1\}$ defined by $b \mapsto \phi(a,b)$. Since $B$ is connected, it follows that $f$ is continuous. So $f$ is constant, and hence $\phi$ is continuous on all sets of the form $\{a\}×B$.

Similarly we can show that $\phi$ is constant on all sets of the form $A×\{b\}$.

Next, fix $(a,b) \in A × B$. Also, let $(a',b') \in A × B$. Then from the results above, we have $\phi(a,b)=\phi(a,b')=\phi(a',b').$

Hence $A×B$ is connected.

Now suppose $(A_i)$ is a countable collection of connected sets for $i\in \mathbb{N}$. We show that the countable Cartesian product $A_1×A_2×...$ is also connected

Clearly, $A_1$ is connected.

Suppose $A_1×A_2×...×A_k$ is connected for some $k\in \mathbb{N}$.

Let $L=A_1×A_2×...×A_k$. Then $L×A_{k+1}$ is connected since the Cartesian product of 2 connected spaces is also connected.

So $A_1×A_2×...×A_k×A_{k+1}$ is connected. Hence our assertion holds by induction.