An exercise on Topology - Analysis
- Tolaso J Kos
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An exercise on Topology - Analysis
Reading a texbook I ran into this cute exercise... which I really liked.. So here goes:
For \( n\geq 1 \) let: $$I_n=\left [ \frac{1}{2n+1},\frac{1}{2n} \right ], \;\;\; S=\bigcup_{n=1}^{\infty}I_n$$ Show that:
a) The set of limit points of \( S \) is \( S \cup \{0 \} \).
b) \( S \) has no isolated points.
c) \( \displaystyle {\rm ext} S=\left ( -\infty, 0 \right )\cup \left [ \bigcup_{n=1}^{\infty}\left ( \frac{1}{2n+2}, \frac{1}{2n+1} \right ) \right ]\cup \left ( \frac{1}{2}, +\infty \right ) \).
For \( n\geq 1 \) let: $$I_n=\left [ \frac{1}{2n+1},\frac{1}{2n} \right ], \;\;\; S=\bigcup_{n=1}^{\infty}I_n$$ Show that:
a) The set of limit points of \( S \) is \( S \cup \{0 \} \).
b) \( S \) has no isolated points.
c) \( \displaystyle {\rm ext} S=\left ( -\infty, 0 \right )\cup \left [ \bigcup_{n=1}^{\infty}\left ( \frac{1}{2n+2}, \frac{1}{2n+1} \right ) \right ]\cup \left ( \frac{1}{2}, +\infty \right ) \).
Imagination is much more important than knowledge.
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