Show that any open subset of R (w std. topology) is a countable union of open intervals.

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 ?

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Expert Solution

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.

Colby Messinger answered 1 year ago

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