topology of real number

kimkibun

consider the set P={1/n:n is counting number}, my classmate said that P is equal to (0,1] but actually i dont agree with him since (0,1] contains irrational numbers. is he correct? also, is it possible for a set not to contain both interior and boundary points?

Number Nine

Can you elaborate? You're correct, of course, that P = {1/n:n is counting number} contains only rational numbers (and not even all of the rational numbers in the unit interval), and so is not equal to (0,1], but maybe he meant something other than equality of sets?

As for your second question: Are you asking if there exists a set that fails to contain both interior an boundary points? In an open set in a metric space, by definition, all points are interior points, so we could take any open ball, for example: every point is interior, and there are no boundary points.