Çarpım Uzaylarında İç

Question

$(X,\tau_1),(Y,\tau_2)$ topolojik uzaylar olmak üzere

$$(A\subseteq X)(B\subseteq Y)\Rightarrow (A\times B)^{\circ}=A^{\circ}\times B^{\circ}$$ olduğunu gösteriniz.

Comments

Bu linkteki bilgiden faydalanabilirsiniz.

Answer 1

$\left.\begin{array}{ccc} (A\subseteq X)(B\subseteq Y)\Rightarrow (A^{\circ}\subseteq A)(B^{\circ}\subseteq B)\Rightarrow A^{\circ}\times B^{\circ}\subseteq A\times B\Rightarrow (A^{\circ}\times B^{\circ})^{\circ}\subseteq (A\times B)^{\circ}\\ \\ (A\subseteq X)(B\subseteq Y)\Rightarrow (A^{\circ}\in\tau_1)(B^{\circ}\in\tau_2)\Rightarrow A^{\circ}\times B^{\circ}\in\tau_1\star\tau_2\Rightarrow (A^{\circ}\times B^{\circ})^{\circ} =A^{\circ}\times B^{\circ} \end{array}\right\}\Rightarrow A^{\circ}\times B^{\circ}\subseteq (A\times B)^{\circ}\ldots (1)$

$\left.\begin{array}{rr} (A\subseteq X)(B\subseteq Y)\Rightarrow A\times B\subseteq X\times Y\Rightarrow (A\times B)^{\circ}\in\tau_1\star\tau_2\\ \\ \pi_1:X\times Y\to X, \pi_1(x,y)=x \,\ (\tau_1\star\tau_2\mbox{ - }\tau_1) \text{ açık}\end{array}\right\}\overset{?}{\Rightarrow} \pi_1[(A\times B)^{\circ}]\in\tau_1\Rightarrow\left(\pi_1[(A\times B)^{\circ}]\right)^{\circ}=\pi_1[(A\times B)^{\circ}]\ldots (2)$

$(A\subseteq X)(B\subseteq Y)\Rightarrow A\times B\subseteq X\times Y\Rightarrow (A\times B)^{\circ}\subseteq A\times B\Rightarrow \pi_1[(A\times B)^{\circ}]\subseteq \pi_1[A\times B]=A\Rightarrow (\pi_1[(A\times B)^{\circ}])^{\circ}\subseteq A^{\circ}\ldots (3)$

$(2),(3)\Rightarrow \pi_1[(A\times B)^{\circ}]\subseteq  A^{\circ}\ldots (4)$

$\left.\begin{array}{rr} (A\subseteq X)(B\subseteq Y)\Rightarrow A\times B\subseteq X\times Y\Rightarrow (A\times B)^{\circ}\in\tau_1\star\tau_2\\ \\ \pi_2:X\times Y\to Y, \pi_2(x,y)=y \,\ (\tau_1\star\tau_2\mbox{ - }\tau_2) \text{ açık}\end{array}\right\}\overset{?}{\Rightarrow} \pi_2[(A\times B)^{\circ}]\in\tau_2\Rightarrow\left(\pi_2[(A\times B)^{\circ}]\right)^{\circ}=\pi_2[(A\times B)^{\circ}]\ldots (5)$

$(A\subseteq X)(B\subseteq Y)\Rightarrow A\times B\subseteq X\times Y\Rightarrow (A\times B)^{\circ}\subseteq A\times B\Rightarrow \pi_2[(A\times B)^{\circ}]\subseteq \pi_2[A\times B]=B\Rightarrow (\pi_2[(A\times B)^{\circ}])^{\circ}\subseteq B^{\circ}\ldots (6)$

$(5),(6)\Rightarrow \pi_2[(A\times B)^{\circ}]\subseteq  B^{\circ}\ldots (7)$

$(4),(7)\Rightarrow (A\times B)^{\circ}\overset{?}{\subseteq} \pi_1[(A\times B)^{\circ}]\times\pi_2[(A\times B)^{\circ}]\subseteq A^{\circ}\times B^{\circ}\ldots (8)$

$$(1),(8)\Rightarrow (A\times B)^{\circ}=A^{\circ}\times B^{\circ}.$$

Not : Son "?" işaretinin gerekçesi yorumdaki linkte mevcut. Diğer "?" işaretlerinin olduğu yerlerde de yine kafa yorulmasının faydalı olacağını düşünüyorum.