In: Advanced Math
Show that any open subset of R (w std. topology) is a countable union of open intervals.
What is the objective of this problem and enough to show ?
Answer: First recall that the standard topology of is the topology generated by the basis
Thus, an open subset in the standard topology is a subset which can be written as union of intervals in
. In other words, for any
there exists
such that
and
Now, we consider the subcollection
Since is countable and the cartessian product of countable sets is countable,
is countable and hence the subcollection
is countable.
Now, we show that an open subset of
can be written as union of open intervals in
. Since
is countable, an open subset
of
can be written as countable union of open intervals. For,
that is, . Now, by the archimedean property of the real numbers, it follows that there exists
such that
and hence there exists
such that
. This shows that
can be written as union of open intervals in
which is countable and hence it follows that an open subset
of
can be written as countable union of open intervals.