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Menge, $\fT = \powerset{X}$ heißt \enquote{diskrete Topologie} \index{Topologie!diskrete} \item $X :=\mdr, \fT_Z := \Set{U \subseteq \mdr | \mdr \setminus U \text{ endlich}} \cup \Set{\emptyset}$ heißt \enquote{Zariski-Topologie} \index{Topologie!Zariski}\\ Beobachtung: $U \in \fT_Z \gdw \exists f \in \mdr[X]$, sodass $\mdr \setminus U = V(f) = \Set{x \in \mdr | f(x) = 0}$ @@ -54,8 +54,8 @@ Es gibt auch Mengen, die weder abgeschlossen, noch offen sind. \begin{definition} Sei $(X, \fT)$ ein topologischer Raum, $M \subseteq X$ eine Teilmenge. \begin{enumerate}[a)] - \item $M^\circ := \Set{x \in M | M \text{ ist Umgebung von } x}$ heißt \textbf{Inneres} oder \textbf{ offener Kern} von $M$. \index{Inneres} \index{Kern!offener} - \item $\displaystyle \overline{M} := \bigcup_{\stackrel{M \subseteq A}{A \text{ abgeschlossen}}} A$ heißt \textbf{abgeschlossene Hülle} oder \textbf{Abschluss} von $M$. \index{Abschluss} + \item $\displaystyle M^\circ := \Set{x \in M | M \text{ ist Umgebung von } x} = \bigcup_{\stackrel{U \subseteq M} {U \in \fT}} U $ heißt \textbf{Inneres} oder \textbf{ offener Kern} von $M$. \index{Inneres} \index{Kern!offener} + \item $\displaystyle \overline{M} := \bigcap_{\stackrel{M \subseteq A}{A \text{ abgeschlossen}}} A$ heißt \textbf{abgeschlossene Hülle} oder \textbf{Abschluss} von $M$. \index{Abschluss} \item $\partial M := \overline{M} \setminus M^\circ$ heißt \textbf{Rand} von $M$. \index{Rand} \item $M$ heißt \textbf{dicht} in $X$, wenn $\overline{M} = X$ ist. \index{dicht} \end{enumerate}