Çarpım Uzaylarında Kompaktlığa Dair

Question

$(X,\tau),(Y,\tau')$ topolojik uzaylar ve $A\subseteq X\times Y$ olmak üzere 

$$(\pi_1[A], \ \tau\text{-kompakt})(\pi_2[A], \ \tau'\text{-kompakt})(A\in \mathcal{C}(X\times Y,\tau\star \tau'))$$

$$\Rightarrow$$

$$A, \ \tau\star\tau'\text{-kompakt}$$ olduğunu gösteriniz.

 

Not: $\mathcal{C}(X,\tau):= \{F|(F\subseteq X)(F, \ \tau\text{-kapalı})\}$

Answer 1

$\left.\begin{array}{rr} \pi_1[A], \ \tau\text{ - kompakt}\overset{?_1}{\Rightarrow} \left(\pi_1[A],\tau_{\pi_1[A]}\right) \text{ kompakt uzay} \\ \\ \pi_2[A], \ \tau'\text{ - kompakt}\Rightarrow \left(\pi_2[A],\tau'_{\pi_2[A]}\right) \text{ kompakt uzay} \end{array}\right\}\overset{?_2}{\Rightarrow}$
  

$\left.\begin{array}{rr} \Rightarrow \left(\pi_1[A]\times \pi_2[A], \tau_{\pi_1[A]}\star\tau'_{\pi_2[A]}\right) \text{ kompakt uzay} \\ \\ \tau_{\pi_1[A]}\star \tau'_{\pi_2[A]}\overset{?_3}{=}(\tau\star \tau')_{\pi_1[A]\times \pi_2[A]}\end{array}\right\}\Rightarrow$

$\left.\begin{array}{rr} \Rightarrow \left(\pi_1[A]\times \pi_2[A], (\tau\star \tau')_{\pi_1[A]\times \pi_2[A]} \right)\text{ kompakt uzay}\\ \\ (A\in \mathcal{C}(X\times Y,\tau\star\tau'))(A\subseteq \pi_1[A]\times\pi_2[A])\end{array}\right\}\Rightarrow$

$\overset{?_4}{\Rightarrow} A, \ (\tau\star \tau')_{\pi_1[A]\times \pi_2[A]}\text{ - kompakt}$

$\overset{?_5}{\Rightarrow} A, \ \tau\star \tau'\text{ - kompakt.}$

Comments

Geniş bir zamanda ispatta iyileştirmeler yapacağım.

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$?_4$ gerekçesi buradaki linkte mevcut.