Let $(X_1,\mathcal O_1)$ and $(X_2, \mathcal O_2)$ be topological spaces.

The product topology is defined as the smallest topology of $X_1 \times X_2$ s.t. $f_1 : X_1 \times X_2 \to X_1$ given by $(x_1, x_2)\mapsto x_1$ and $f_2 : X_1 \times X_2 \to X_2$ given by $(x_1, x_2)\mapsto x_2$ are continuous.

Let this product topology $\mathcal O.$

Then, I want to show $\mathcal B:=\{O_1\times O_2 \mid O_1\in \mathcal O_1, O_2\in \mathcal O_2 \}$ is the base of $\mathcal O$.

What I have to show are

(i) $\mathcal B\subset \mathcal O.$

(ii) For all $O\in \mathcal O$, there exist an index set $\Lambda$ and the family of open sets $\{V_\lambda \mid \lambda \in \Lambda \}\subset \mathcal B$ s.t. $O=\bigcup_{\lambda \in \Lambda} V_\lambda.$

My proof for (i) is here.

Let $O\in \mathcal B.$

There exists $O_1\in \mathcal O_1, O_2\in \mathcal O_2$ s.t. $O=O_1\times O_2$.

Since $\mathcal O$ is a product topology, $f_1, f_2$ are continuous, and now, $O_1$ is open in $X_1$ and $O_2$ is open in $X_2$, so $f_1^{-1}(O_1)$ and $f_2^{-1}(O_2)$ is open in $X_1\times X_2,$ i.e., $f_1^{-1}(O_1)$, $f_2^{-1}(O_2)\in \mathcal O.$

Then, \begin{align} f_1^{-1}(O_1)&=\{(x_1,x_2)\in X_1\times X_2 \mid f_1(x_1, x_2)\in O_1\} \\ &=\{(x_1,x_2)\in X_1\times X_2 \mid x_1 \in O_1\} \\ &=\{(x_1,x_2)\in O_1\times X_2\} \\ &=O_1\times X_2. \end{align}

Similarly, I can see $f_2^{-1}(O_2)=X_1\times O_2.$

And $f_1^{-1}(O_1)\cap f_2^{-1}(O_2)=(O_1\times X_2)\cap (X_1\times O_2)=O_1\times O_2=O$.

Thus $O=f_1^{-1}(O_1)\cap f_2^{-1}(O_2)\in \mathcal O.$

Therefore $\mathcal B\subset \mathcal O.$

I have difficulty in proving (ii).

Of course, at first, let $O\in \mathcal O$.

But I don't know how I should find an index set $\Lambda$ and the family of open sets $\{V_\lambda \mid \lambda \in \Lambda \}\subset \mathcal B$ s.t. $O=\bigcup_{\lambda \in \Lambda} V_\lambda.$ I think I have to use the propertioes of $\mathcal O$ (the milimality and the continuity of $f_1, f_2$ and so on) but I don't know how.

I'd like you to give me any ideas.


Hint: Consider the collection of all sets $O \subset X_1\times X_2$ which can be written as a union of sets from $\mathcal B$. Check that this defines a topology and $f_1,f_2$ are continuous for this topology. By definition of product topology it follows that every set in the product topology belongs to this topology and that is what you want to prove. [You will need the following: $(\cup_i A_n) \cap (\cup_j B_j)=\cup_{i,j} (A_i\cap B_j)$ and intersection of two sets in $\mathcal B$ belngs to $\mathcal B$].


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