Interior of direct product (Cartesian product) is equal to direct product of their interiors.

Question

Let $A \subset \mathbb{R}^n, B \subset \mathbb{R}^m.$

I want to prove $\mathrm{int}(A\times B)=(\mathrm{int}A)\times (\mathrm{int} B)$.

I could prove $\mathrm{int}(A\times B)\supset (\mathrm{int}A)\times (\mathrm{int} B)$, but I'm difficulty in proving the reverse.

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Let $x\in \mathrm{int}(A\times B)$.

Due to the definition of int$(A\times B),$ there exists an open set $W$ s.t. $x\in W\subset A\times B.$

Since $x\in W\subset A\times B,$ I can write $W=U\times V$, where $U\subset A, V\subset B$ and $x=(u,v)$ where $u\in U, v\in V.$

So, all that is left is to show $U,V$ are open.

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Here are two questions of mine.

(i) How can I show $U,V$ are open ?

(ii) Is $ \big[ W\subset A\times B \Rightarrow W$ is written as $W=U\times V\big]$ in my proof correct ?

Answer 1

$\operatorname{Int}(A) \times \operatorname{Int}(B)$ is open and a subset of $A \times B$. As the interior is the largest open subset of a set, $\operatorname{Int}(A) \times \operatorname{Int}(B) \subseteq \operatorname{Int}(A \times B)$ is immediate. Also, as projections are open, $\pi_X[\operatorname{Int}(A \times B)]$ is an open subset of $A$ and by the same maximality $\pi_X[\operatorname{Int}(A \times B)] \subseteq \operatorname{Int}(A)$ and likewise $\pi_Y[\operatorname{Int}(A \times B)]$ is an open subset of $B$ from which $\pi_Y[\operatorname{Int}(A \times B)] \subseteq \operatorname{Int}(B)$ follows. Simple set theory then tell us $\operatorname{Int}(A \times B) \subseteq \operatorname{Int}(A) \times \operatorname{Int}(B)$, finishing the proof.

Answer 2

If $(x,y)\in \mbox{Int} (A\times B ) $ then there exists an open set $W\subset A\times B$ such that $(x,y) \in W.$ But by the definition of product topology there exists a open sets $U,V$ such that $(x,y)\in U\times V\subset W\subset A\times B$ hence $x\in \mbox{Int} (A)$ and $y\in \mbox{Int} ( B )$ and therefore $(x,y)\in \mbox{Int} (A)\times \mbox{Int} ( B ).$ And consequently $$\mbox{Int} (A\times B )\subset \mbox{Int} (A)\times \mbox{Int} ( B ).$$