Answer to Question #217093 in Differential Geometry | Topology for Prathibha Rose

Question

Prove that a countable product of connected space is connected

Answer 1

Solution

Let A and B be connected spaces such that a map

F:A×B$\to$ {0,1}

To prove that A×B is connected we have to show that F is a constant.

Let

$(a_1,b_1)\in A×B$

Now ,we define

g:A$\to$ {0,1}

g:a$\to$ f(a,b₁)

and

h:B$\to$ {0,1}

h:b$\to$ f(a₁,b)

Since A and B are connected, h and g must be constant, hence

f(a,b₁)=f(a₁,b)=f(a₁,b₁) for all (a,b)$\in$ A×B

If we fix some other;

$(a_2,b_2)\in A×B$

But $(a_1,b_1)\ne (a_2,b_2)$

We obtain

f(a,b₂)=f(a₂,b)=f(a₂,b₂) for all (a,b)$\in$ A×B

In

f(a₁,b)=f(a₁,b₁), take b=b₂ ,we have

f(a₁,b₁)=f(a₁,b₂)

In

f(a,b₂)=f(a₂,b₂), take a=a₁, we have

f(a₁,b₂)=f(a₂,b₂)

Hence

f(a₁,b₁)=f(a₂,b₂)

Since

(a₁,b₁) and (a₂,b₂) are arbitrary;

f(a₁,b₁)=f(a₂,b₂) for all (a₁,b₁),(a₂,b₂) $\in$ A×B

Therefore, F is a constant function.

Hence A×B is connected.