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\title{{Regularity of atomic and molecular Coulombic
    eigenfunctions and associated electron densities}}



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\noindent{\bf 4. Tutorium zu MPIIA}  \hspace*{\fill}{09.05.-12.05.2005}


\noindent{\bf Aufgabe 9:} a) Sei \((X,d)\) metrischer Raum, 
\(K_r(x):=\{y\in X\,|\, d(x,y)<r\}\). Zeigen Sie, da\ss\ \(K_r(x)\)
offen ist.\\\noindent
b) Sei \(A_r(x):=\{y\in X\,|\, d(x,y)\leq r\}\). Zeigen Sie, durch ein
Gegenbeispiel, da\ss\ \(\overline{K_r(x)}=A_r(x)\) {\it nich}t immer gilt;
welche Inklusion gilt immer?

\noindent{\bf Aufgabe 10:} Sei \(d:\mathbb R^2\times \mathbb
R^2\to\mathbb R\) die Abbildung
\begin{align*}
  d(x,y):=\begin{cases}
      |x|+|y| & \text{ f\"ur }x\neq y\\
      0 & \text{ f\"ur }x=y
\end{cases}
\end{align*}
wo
\(|\,\cdot\,|\) die Euklidische L\"ange ist. Zeigen Sie, da\ss\ \(d\)
eine Metrik ist. (Die {\it Postamt-Metrik}).

\noindent{\bf Aufgabe 11:} Sei \((M,d)\) metrischer Raum, und
\(A\subset M, x\in M\). Sei 
\begin{align*}\dist(x,A):=\inf\{d(x,y)\,|\, y\in A\}.
\end{align*}
Zeigen Sie, da\ss\ \(\dist(\,\cdot\,,A):M\to\mathbb R\) stetig ist.

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