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\begin{document}
\title{Analiz ara s\i nav\i}
\author{David Pierce, MSGS\"U}
\date{28 Mart 2012}
\maketitle\thispagestyle{empty}

\emph{$f\colon\N\to\R^2$ olsun.  $\R^2$ d\"uzleminin topolojisi, \"Oklid
topolojisi olsun.
\begin{itemize}
\item
$\tau$, $\N$ \"uzerinde ayr\i k topoloji olsun (yani her altk\"ume a\c
  c\i k olsun).
\item 
$\sigma$, $\N$ \"uzerinde Fr\'echet topolojisi olsun (yani a\c c\i k
  k\"umeler, t\"umleyenleri sonlu olan k\"umeler olsun).
\end{itemize}}

\begin{problem} 
$\R^2\setminus f[\N]$ her zaman a\c c\i k m\i d\i r?
\end{problem} 

\begin{solution}
Hay\i r.  $f(n)=(1/n,0)$ ise $\R^2\setminus f[\N]$ a\c c\i k de\u
gildir (\c c\"unk\"u $(0,0)\in\R^2\setminus f[\N]$ ama bu k\"ume, bu
noktan\i n kom\c sulu\u gu de\u gildir).
\end{solution}

\begin{problem}
$\R^2\setminus f[\N]$ a\c c\i k olabilir mi?
\end{problem} 

\begin{solution}
Evet.  $f(n)=(0,0)$ ise $\R^2\setminus f[\N]$ a\c c\i kt\i r.
\end{solution}

\begin{remark}
  Bu soru i\c cin Al\i\c st\i rma 4.17 faydal\i d\i r.
\end{remark}


\begin{problem}
$\tau$'ya g\"ore $f$ her zaman s\"urekli midir?
\end{problem} 

\begin{solution}
Evet, \c c\"unk\"u $\tau$'ya g\"ore $f\inv[V]$ her zaman a\c c\i kt\i r.
\end{solution}


\begin{problem}
$\tau$'ya g\"ore $f$ s\"urekli olabilir mi?
\end{problem} 

\begin{solution}
Evet; 3'e g\"ore $n\mapsto(0,0)$ s\"ureklidir (\c c\"unk\"u bu
fonksiyon $\N$'den $\R^2$ d\"uzlemine gider: bu yeter).
\end{solution}


\begin{problem}
$\sigma$'ya g\"ore $f$ her zaman s\"urekli midir?
\end{problem} 

\begin{solution}
Hay\i r; $\sigma$'ya g\"ore $n\mapsto(x,0)$ s\"urekli de\u gildir (\c
c\"unk\"u bu fonksiyon alt\i nda $B((1,0),1)$ a\c c\i k topunun \"onimgesi
$\{1\}$'dir, ve bu k\"ume ne bo\c s ne t\"umleyeni sonlu olan k\"umedir).
\end{solution}


\begin{problem}
$\sigma$'ya g\"ore $f$ s\"urekli olabilir mi?
\end{problem}

\begin{solution}
Evet; $n\mapsto(0,0)$ s\"ureklidir (\c c\"unk\"u her sabit fonksiyon
s\"ureklidir).  
\end{solution}

\begin{remark}
  Her \c c\"oz\"umde $(x,y)$ ifadesi, aral\i k de\u gil, $\R^2$
  d\"uzlemde noktad\i r.
\end{remark}

\end{document}