Indeed, there exists a very famous closed set called the Cantor set whose structure is much more interesting. To … Give an example of a countable collection of disjoint open intervals.
For suppose that C is any in nite collection of disjoint open intervals. This must be a countable set, so we can enumerate it: let.
Between any two closed intervals, there would have to be more closed intervals. Part (b) Give an example of an uncountable collection of disjoint open intervals, or argue that no such collection exists. Write $a\sim b$ if the closed interval $[a, b]$ or $[b, a]$ if $b
The collection f(n;n+ 1) : n2Ngsu ces.
No such collection exists. A set B is called a G set if it can be written as the countable intersection of open sets. $G$ is therefore the union of disjoint equivalence classes. The following link shows that opne set can is countable disjoint union of open intervals: Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals.
kathrynmath said: A set A is called a F set if it can be written as the countable union of closed sets. Though every open set in R is a disjoint union of countably many open intervals, it is not true that every closed set is a disjoint union of closed intervals.
Therefore any set of disjoint intervals of strictly positive length must be countable, since we can inject any such set into. 2) Let Ix be the smallest neighborhood of x. Consider the sequence of sets. a) Show that a closed interval [a,b] is a G set