Using the fact that a subset of R is closed in R if it conta

Using the fact that a subset of R is closed in R if it contains all of its accumulation points. Explain why a. Q is not closed in R b. N and Z are closed in R

Solution

a)

None of the points in Q are interior points. Let x Q be a rational number. Recall from Real Analysis that between any two real numbers there exists an irrational number. Thus every open interval (xr, x+r) (i.e. every open ball in R with centre x) contains an irrational number. This shows x is not an interior point of Q.

b)

RZ=_nZ(n,n+1) is a union of open sets and therefore open. Since the complement of Z is open in RR, Z is closed in R.

we can prove that N is a closed set through the use of the compliment of N. Since R\\ N is open, N must be closed.