Let S = [0,4]. Show S is closed.
How can I show that the bd S is a subset of S, or that R/S is open?
I know that simply stating the bd should be enough to say that S is closed, but my professor wants us to be able to also show bd{0 , 4}.
I know that if it were (0,4) I could say that (0 - E, 0+ E) intersected with S and R/S is non empty, but is it sufficient to say that if S is closed then the bd S = bd (R/S) so R/S is empty?
Also at what point when x is in bd does (x - E, x + E) intersected with S and R/S non empty? This would only be true for an open set correct?
An alternative way to do this (which is just implicitly noting that is open) is to suppose that . We have two cases, either in which case note that ; or in which case . And clearly and . If the first case is two then choosing shows that and if the second is true then choosing shows that . In either case is an interior point of . Thus is open which implies that is closed.