Show that a subset of R is open if an only if it can be expressed...

Show that a subset of R is open if an only if it can be expressed as a union of open intervals

Expert Solution

let A be an open set in , then every point in A is an interior point, that is, . where are open intervals in .

now and any point in A is in one of such open intervals in that is . therefore . which shows that any open set can be expressed as union of open intervals.

The converse part is trivial as open intervals are also open sets in and union of open sets in is again an open set. Therefore union of open intervals in is an open set.

Colby Messinger answered 2 years ago

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