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Dann gilt + \[ + A \text{ abgeschlossen} + \iff + \text{Ist } \left(x^{(k)}\right)_{k\in\N} \text{ konvergente Folge in } A + \text{ mit } \lim_{k \to \infty} x^{(k)} = a\text{, dann } a \in A + .\] +\end{satz} + +\begin{proof} + \begin{itemize} + \item ,,$\implies$'': Sei $A$ abgeschlossen und $\left( x^{(k)} \right)_{k \in \N}$ + konvergente Folge in $A$ mit + \[ + \lim_{k \to \infty} x^{(k)} = x + .\] + Ang.: $x \not\in A$, d.h. $x \in A^{C}$. Da $A^{C}$ offen, folgt, es ex. + ein $\epsilon > 0$, s.d. $K_{\epsilon}(x) \subset A^{C}$. + Mit $x = \displaystyle \lim_{k \to \infty} x^{(k)}$ folgt, dass fast alle + Folgenelemente $x^{(k)}$ in $K_{\epsilon}(x) \subset A^{C}$ liegen. + + Widerspruch zu: $\left( x^{(k)} \right)_{k \in \N} \subset A $. Damit folgt + $x \in A$. + \item ,,$\impliedby$'': Sei $A \subset \mathbb{K}^{n}$ s.d. alle konvergenten + Folgen in $A$ einen Grenzwert in $A$ haben. + + Zu zeigen: $A^{C}$ offen. Sei $x \in A^{C}$ beliebig. Dann g.z.z.: $\exists \epsilon > 0$ + s.d. $K_{\epsilon}(x) \subset A^{C}$. + + Ang.: $A^{C}$ nicht offen. Dann ex. $\forall k \in \N$ ein Punkt $x^{(k)}$ + mit $x^{(k)} \in A \cap K_{\frac{1}{k}}(x)$. Dann ist $x^{(k)} \in A$ + $\forall k \in \N$ und $\Vert x - x^{(k)} \Vert \le \frac{1}{k}$. Damit folgt + \[ + x^{(k)} \xrightarrow{k \to \infty} x \stackrel{\text{Vorr.}}{\implies} x \in A \quad \contr + \implies A^{C} \text{ offen } \implies A \text{ abgeschlossen} + .\] + \end{itemize} +\end{proof} + +\begin{definition}[Randpunkt] + Sei $M \subset \mathbb{K}^{n}$ eine Teilmenge. Ein Punkt $a \in \mathbb{K}^{n}$ heißt + Randpunkt von $M$, falls in jeder Umgebung von $a$ sowohl ein Punkt von $M$, als auch + ein Punkt von $M^{C} = \mathbb{K}^{n} \setminus M$ liegt. + + Die Menge aller Randpunkte von $M$ heißt der Rand von $M$, bezeichnet mit $\partial M$. +\end{definition} + +\begin{figure}[h!] + \begin{tikzpicture}[scale=2] + \draw plot [smooth cycle] coordinates {(0,0) (1,1) (2,1) (3, 2) (3,0.5)}; + \draw (1,1) circle [radius=0.15cm]; + \draw[fill=black] (1,1) circle [radius=0.02cm]; + \end{tikzpicture} + \centering + \caption{Randpunkt einer Menge $M \subset \mathbb{K}^{n}$} +\end{figure} + +\begin{bsp} + \begin{enumerate}[(1)] + \item Für $I \in \{ [a,b[ \;, [a,b], \;]a,b], \;]a,b[\;\} $ gilt + $\partial I = \{a, b\}$. + + $\partial [a, \infty[ \; = \{a\}$\\ + $\partial ]a, \infty[ \; = \{a\}$ + \item Für $K_1(0)$ gilt + \begin{align*} + \partial K_1(0) &= \partial \{x \in \R^{n} \mid \Vert x \Vert < 1\} \\ + &= \;\; \{ x \in \R^{n} \mid \Vert x \Vert = 1 \} \\ + & \quad \quad \text{,,Einheitssphäre''} + .\end{align*} + \item $\Q \subset \R$, $\partial \Q = \R$, weil in jeder Umgebung eines Punktes in + $\Q$, gibt es rationale und irrationale Zahlen. Der Rand von $\R$ ist leer. + \end{enumerate} +\end{bsp} + +\begin{definition}[Inneres, Abschluss] + Sei $M \subset \mathbb{K}^{n}$ + \begin{itemize} + \item Die Menge $M^{\circ} := M \setminus \partial M$ heißt das + \underline{Innere} von $M$. + \item Die Menge $\overline{M} := M \cup \partial M$ heißt + der \underline{Abschluss} von $M$. + \end{itemize} +\end{definition} + +\begin{satz}[Inneres ist Offen, Abschluss ist abgeschlossen] + Sei $M \subset \mathbb{K}^{n}$. + \begin{enumerate}[(i)] + \item Die Menge $M^{\circ} = M \setminus \partial M$ ist offen. + $M^{\circ}$ ist die größte offene Menge in $M$. + \item Die Menge $\overline{M} = M \cup \partial M$ ist abgeschlossen. + $\overline{M}$ ist die kleinste abgeschlossene Menge, die $M$ umfasst. + \item Der Rand $\partial M$ ist abgeschlossen. + \end{enumerate} +\end{satz} + +\begin{proof} + \begin{enumerate}[(i)] + \item Z.z.: $M \setminus \partial M$ offen. + + Sei $x \in M \setminus \partial M$ beliebig, dann ex. $\epsilon > 0$, s.d. + $K_{\epsilon}(x) \subset M$ $(\implies K_{\epsilon}(x) \cap M^{C} = \emptyset)$, sonst + wäre $x \in \partial M$. + + Für dieses $\epsilon$ gilt auch + $K_{\epsilon}(x) \cap \partial M = \emptyset$, denn + falls $z \in K_{\epsilon}(x) \cap \partial M$ existiert, dann ist + $K_{\epsilon}(x)$ Umgebung von $z$ und folglich + $K_{\epsilon}(x) \cap M^{C} \neq \emptyset$. + + Damit folgt: + \[ + K_{\epsilon}(x) \subset M \setminus \partial M \implies M \setminus \partial M \text{ offen} + .\] + + Sei $U \subset M$ offen, dann ist analog $U \cap \partial M = \emptyset$. Damit gilt + $U \subset M \setminus \partial M$. Da $U$ beliebig, folgt damit + $M \setminus \partial M =: M^{\circ}$ ist größte offene Teilmenge von $M$. + \item Z.z.: $M \cup \partial M$ abgeschlossen. + + Betrachte $M^{C} = \mathbb{K}^{n} \setminus M$. Nach Definition des Rands + gilt $\partial M^{C} = \partial M$. Damit folgt mit (i), dass + $M^{C} \setminus \underbrace{\partial M}_{= \partial M^{C}}$ offen ist. Dann + \[ + (M^{C} \setminus \partial M)^{C} + = \mathbb{K}^{n} \setminus (M^{C} \setminus \partial M) + = \underbrace{(\mathbb{K}^{n} \setminus M^{C})}_{= M} \cup \partial M = M \cup \partial M + .\] D.h. $M \cup \partial M$ ist abgeschlossen. + + Sei $V \in K^{n}$ abgeschlossen mit $M \subset V$. Dann gilt + $V^{C}$ ist offen und $V^{C} \subset M^{C}$. Damit folgt mit (i): + \[ + \underbrace{V^{C}}_{\text{offen}} + \subset M^{C} \setminus \underbrace{\partial M^{C}}_{=\partial M} = M^{C} \setminus \partial M + \implies \mathbb{K}^{n} \setminus (M^{C} \setminus \partial M) + = (M \cup \partial M) \subset V + .\] + Da $V$ beliebig, folgt damit $M \cup \partial M$ ist kleinste abgeschlossene Menge, die + $M$ umfasst. + \item Mit $\partial M = (M \cup \partial M) \setminus (M \setminus \partial M)$ folgt + \[ + \mathbb{K}^{n} \setminus \partial M + = \underbrace{\left( \mathbb{K}^{n} \setminus (M \cup \partial M)\right)}_{\text{offen}} + \cup \underbrace{(M \setminus \partial M)}_{\text{offen}} + .\] Damit ist $\mathbb{K}^{n} \setminus \partial M$ offen, also $\partial M$ abgeschlossen. + \end{enumerate} +\end{proof} + +\begin{definition}[Kompaktheit] + Eine Menge $M \subset \mathbb{K}^{n}$ heißt \underline{kompakt} + (bzw. \underline{folgenkompakt}), wenn jede Folge aus $M$ eine + konvergente Teilfolge mit Grenzwert in $M$ besitzt. +\end{definition} + +\begin{bsp} + \begin{enumerate}[(i)] + \item Sei + \[ + \left( x^{(k)}\right)_{k \in \N} \subset \mathbb{K}^{n}, x^{(k)} \xrightarrow{k \to \infty} x + .\] Dann ist $A := \{ x^{(k)} \mid k \in \N\} \cup {x}$ kompakt. + \item $]0,1[$ ist nicht kompakt, denn $\left( \frac{1}{2k} \right)_{k \in \N} \subset ]0,1[$, + $\frac{1}{2k} \xrightarrow{k \to \infty} 0$. + + Auch: $\left( 1 - \frac{2}{k} \right)_{k \in \N} \subset ]0,1[$, + $1 - \frac{1}{2k} \xrightarrow{k \to \infty} 1$ + \end{enumerate} +\end{bsp} + +\begin{definition}[Überdeckung] + Eine Familie $(U_i)_{i\in I}$ von Teilmengen $U_i \subset \mathbb{K}^{n}$ heißt + Überdeckung von $M$, falls gilt + \[ + M \subset \bigcup_{i \in I} U_i + .\] Eine Überdeckung heißt offen bzw. abgeschlossen, wenn alle $U_i$ offen bzw. abgeschlossen sind. +\end{definition} + +\begin{satz}[Charakterisierung von Kompaktheit] + Sei $M \subset \mathbb{K}^{n}$ eine Teilmenge. Dann sind + die folgenden Aussagen äquivalent: + \begin{enumerate}[(i)] + \item $M$ ist folgenkompakt + \item $M$ ist beschränkt und abgeschlossen + \item Jede offene Überdeckung $\left( U_i \right)_{i \in I} $ von $M$ enthält + eine \underline{endliche} Überdeckung von $M$, d.h. es existieren endlich + viele Indizes $i_1, \ldots, i_k \in I$, s.d. $M \subset (U_{i_1} \cup \ldots \cup U_{i_k})$ + (sogenannte Überdeckungseigenschaft von Heine und Borel). + \end{enumerate} + \label{satz:charakter-kompaktheit} +\end{satz} + +\begin{proof} + \begin{itemize} + \item (i) $\implies$ (ii): Sei $M \subset \mathbb{K}^{n}$ folgenkompakt. Dann + existieren für alle konvergenten Folgen $\left( x^{(k)} \right)_{k \in \N} \subset M$ + eine konvergente Teilfolge mit Grenzwert in $M$. Damit liegt + auch der Grenzwert von $\left( x^{(k)} \right)_{k \in \N}$ in $M$. + Also ist $M$ abgeschlossen. + + Ang.: $M$ ist nicht beschränkt. Dann ex. eine Folge + $\left( x^{(k)} \right)_{k \in \N}$ mit $\Vert x^{(k)} \Vert \xrightarrow{n \to \infty} \infty$. + Damit hat $\left( x^{(k)} \right)_{k \in \N}$ keine konvergente Teilfolge. + Widerspruch zur Kompaktheit von $M$. Also ist $M$ beschränkt. + \item (ii) $\implies$ (i): Sei $M \subset \mathbb{K}^{n}$ beschränkt und + abgeschlossen. Dann folgt mit \ref{satz:bolzano}, dass alle Folgen + $\left( x^{(k)} \right)_{k \in \N} \subset M $ beschränkt sind und eine + konvergente Teilfolge $\left( x^{(k_j)} \right)_{j \in \N} \xrightarrow{j \to \infty} x$ + besitzen. Da $M$ abgeschlossen ist, folgt $x \in M$. + + Also ist $M$ folgenkompakt. + \item (iii) $\implies$ (i): Sei $M \subset \mathbb{K}^{n}$ und $M$ besitze die + Überdeckungseigenschaft. Sei weiter $\left( x^{(k)} \right)_{k \in \N} \subset M $ beliebig. + + Z.z.: Es ex. eine konvergente Teilfolge $\left( x^{(k_j)} \right)_{j \in \N}$ + mit $x^{(k_j)} \xrightarrow{j \to \infty} x \in M$. + + Ang.: Solche Teilfolge existiert nicht. Dann gilt: $\forall x \in M$ existiert + eine offene Umgebung $U_x$ von $x$, die nur endlich viele + Folgenelemente von $\left( x^{(k)} \right) $ enthält (wären in jeder Umgebung + von $x$ unendlich viele Folgenelemente, dann existiert eine konvergente Teilfolge). + + Damit ist $M = \bigcup_{x \in M} U_x$ eine offene Überdeckung, d.h. es existiert + nach Vorr. eine endliche Überdeckung von $M$, d.h. eine endliche + Menge $I$ mit + \[ + \{x_i \mid x_i \in M, i \in I\} =: M_i \text{ s.d. } + M \subset \bigcup_{x_i \in M_i} U_{x_i} + .\] Da $\forall i \in I$ $U_{x_i}$ nur endlich viele Folgenelemente enthält, + enthält $M$ endlich viele Folgenelemente von $\left( x^{(k)} \right)_{k \in \N}$, d.h. + $\left( x^{(k)} \right)_{k \in \N} \not\subset M$ $\contr$. + + Also existiert eine Teilfolge $\left( x^{(k_j)} \right)_{j \in \N}$ mit + $x^{(k_j)} \xrightarrow{k \to \infty} x \in M$ + \item (ii) $\implies$ (iii): Sei $M$ beschränkt und abgeschlossen und sei + $\{U_i, \in I\} $ eine offene Überdeckung von $M$. + + Zu zeigen: Es existiert eine endliche Überdeckung von $M$. + + Ang.: Eine solche Überdeckung existiert nicht. Konstruiere induktiv eine Folge + von beschränkten, abgeschlossenen Würfeln in $\mathbb{K}^{n}$: + \[ + Q_0 \supset Q_1 \supset Q_2 \supset \ldots + .\] mit + \begin{enumerate}[(1)] + \item $M \cap Q_i$ wird nicht durch endlich viele $U_{i_k}$ überdeckt. + \item Kantenlänge von $Q_m = 2^{-m}$ Kantenlänge von $Q_0$. + \end{enumerate} + Sei $Q$ beschränkter abgeschlossener Würfel in $\mathbb{K}^{n}$ mit + Kantenlänge $L$, s.d. $M \subset \Q$. + \begin{figure}[h!] + \begin{tikzpicture}[scale=0.2] + \draw (0,0) -- (0,10) -- (10,10) -- (10,0) -- (0,0); + \draw plot [smooth cycle] coordinates {(2, 3) (2,7) (8,8) (8, 2)}; + \node at (12, 5) {$L$}; + \node at (5, -2) {$L$}; + \node at (6, 6) {$M$}; + \end{tikzpicture} + \centering + \caption{Abgeschlossener Würfel $Q \subset K^{n}$ mit Kantenlänge $L$ und $M \subset Q$} + \end{figure} + Setze $Q_0 = Q$, Kantenlänge von $Q_0 = L$. Sei $Q_m$ bereits konstruiert. Sei + \[ + Q_m = I_1 \times I_2 \times \ldots \times I_n + .\] Länge $(I_k)$ = Kantenlänge $(Q_m)$ $\forall k$ = $2^{-m} L$ + + Wir zerlegen jedes $I_i$ in 2 abgeschlossene Intervalle mit halber Länge + $I_i^{(1)}$ und $I_i(^{2)}$ und setzen für $(s_1, \ldots, s_n) \in \{1, 2\}^{n}$ + \[ + Q_m^{s_1, \ldots, s_n} := I_1^{(s_1)} \times \ldots \times I_n^{(s_n)} + .\] Wir erhalten $2^{n}$ Würfel mit + \[ + Q_m := \bigcup_{(s_1, \ldots, s_n) \in \{1, 2\}^{n}} Q_m^{(s_1, \ldots, s_n)} + .\] Da $M \cap Q_m$ nicht von endlich vielen $U_{i_k}$ überdeckt wird, gilt dies + auch für einen Würfel + \[ + Q_{m+1} := Q_m^{(s_1, \ldots, s_n)} + .\] Es gilt für die Kantenlänge $(Q_{m+1})$ = $\frac{1}{2}$ Kantenlänge $(Q_m)$ = $2^{-(m+1)} L$. + + Für $k \in \N$ wähle $x^{(k)} \in Q_k \cap M$. Damit ist $\left( x^{(k)} \right)_{k \in \N}$ eine + Cauchy-Folge in $\mathbb{K}^{n}$, da nach Konstruktion von $Q_1, Q_2, \ldots$ + \[ + \Vert x^{(l)} - x^{(k)} \Vert \le 2^{-n_0} L, \quad \forall l, k \ge n_0 + .\] + Damit folgt $x^{(k)} \xrightarrow{k \to \infty} x \in M$ und + $x \in \bigcup_{i \in I} U_i$, weil $M \subset \bigcup_{i \in I} U_i$. Also + existiert ein $i_k$, s.d. $x \in U_{i_k}$ liegt. Damit liegen fast alle + $Q_m$ in $U_{i_k}$. Das heißt fast alle $M \cap Q_m$ liegen + in $U_{i_k}$. Widerspruch zur Annahme, dass eine endliche Überdeckung nicht existiert. + + Also existiert eine endliche Überdeckung von $M$. + \end{itemize} +\end{proof} + +\begin{bem} + Wichtige Voraussetzung für die Überdeckungseigenschaft von Heine und Borel ist, dass + $\mathbb{K}^{n}$ \underline{endlich}-dimensional ist. + + In unendlich dimensionalen Banach-Räumen wie z.B.: $C[a,b]$ ist dies nicht möglich. +\end{bem} + +\begin{korrolar} + Jede abgeschlossene Teilmenge einer kompakten Menge in $\mathbb{K}^{n}$ ist + ebenfalls kompakt. +\end{korrolar} + +\begin{proof} + Sei $M \subset \mathbb{K}^{n}$ kompakt und $A \subset M$ abgeschlossen. Wegen + \ref{satz:charakter-kompaktheit} ist $M$ beschränkt. Damit ist auch $A \subset M$ beschränkt + und somit nach \ref{satz:charakter-kompaktheit} kompakt. +\end{proof} + +\end{document}