Show that any open set in Rd can be written as the countable

Show that any open set in R^d can be written as the countable disjoint union of half-open rectangles.

Solution

any open set is a countable union of basic open sets which are open rectangles.

and each open rectangle can be written as a countable union of half open rectangles.

And Since countable union of counable set is countable.

any open set is therefore can be wriiten as countable union of half open rectangles