Answer to Question #188120 in Real Analysis for Nikhil

Question

Check, whether the collection of G, given by

G'= { ] 1/n+1,1/n [ : n∈N } is a open curve of ]0,1[

Answer 1

$G={(5-n,5+\dfrac{1}{n})|n\in N}$

Let $B_n=(5-n,5+\dfrac{1}{n})$

Then,$B_1=(4,6) ,B_2=(3,5+\dfrac{1}{2})$

   $B_3=(2,5+\dfrac{1}{3}), B_4=(1,5+\dfrac{1}{4})$

   $B_5=(0,5+\dfrac{1}{5}),.....$

Here $B_2 B_1,B_3 B_2,...., B_{n+1} B_n$

Hence the collections of sets {$B_n$ } is notnested.

Also, we see that, {$B_n$ } are not mutually disjoint a$B_1 \cup B_2\neq \phi, B_2\cap B_3\neq \phi,..., B_n\cap B_{n+1}\neq \phi$

Hence The given set G is not an open curve on [0,1]