This is an exercise from a topological book. It is this:
Show that any open subsets of the real line are $F_\sigma$-sets.
Could anybody help to solve it?
This is an exercise from a topological book. It is this:
Show that any open subsets of the real line are $F_\sigma$-sets.
Could anybody help to solve it?
First note that open subsets of the real line are countable unions of (pairwise disjoint) open intervals. Next show that every open interval is an F$_\sigma$-set. As countable unions of F$_\sigma$-sets are also F$_\sigma$, this gives the result.
Hint: start with an open interval. Can you show it is the union of countably many closed intervals?