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\theoremstyle{definition}
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\begin{document}
\title{Analiz k\i sa s\i nav\i\ 8}
\author{David Pierce, MSGS\"U}
\date{25 Nisan 2012}
\maketitle\thispagestyle{empty}
%{\large \textbf{Analiz k\i sa s\i nav\i} (David Pierce), 20 \c Subat 2012}


\begin{problem}
Her say\i lamaz metrik uzay\i nda, bir $a$ noktas\i\ i\c cin, bir $r$
yar\i\c cap\i\ i\c cin, $\mathrm B(a;r)$ a\c c\i k topunun say\i lamaz
oldu\u gunu g\"osterin. 
\end{problem}

\begin{solution}
  Her metrik uzay, $\bigcup_{n\in\N}\ball an$ bile\c simine e\c
  sittir.  Her $\ball an$ topu say\i labilirse, t\"um uzay da say\i
  labilir. 
\end{solution}

\begin{problem}
Her Boole halkas\i n\i n $xy=\inf\{x,y\}$ e\c sitli\u gini sa\u
glad\i\u g\i n\i\ g\"osterin. 
\end{problem}

\begin{solution}
  $xy\leq x$ \c c\"unk\"u $xyx=x^2y=xy$.  Ayn\i\ \c sekilde $xy\leq
  y$.  \c Simdi $z\leq x$ ve $z\leq y$ e\c sitsizliklerini varsayal\i
  m.  Bu durumda
  \begin{align*}
    zx&=z,&zy&=z.
  \end{align*}
O zaman $zxy=zy=z$, yani $z\leq xy$.  \"Oyleyse $xy$, $\{x,y\}$
k\"umesinin en b\"uy\"uk alt s\i n\i r\i d\i r.
\end{solution}

\begin{remark}
  Verilmi\c s Boole halkas\i, $(\pow{\Omega},\triangle,\cap)$
  bi\c cimindeyse, \"onerme apa\c c\i kt\i r.  Ama her Boole halkas\i,
  bu bi\c cimde de\u gildir.
\end{remark}

\end{document}
