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**Kiefer** Proove each of the following:

(1) For any point p an element of R and N subset of R, N is a neighborhood for p in R* iff p is an element of N.

R represents the real numbers

(2) For any point p an element of R, N is a neighborhood for p in R_* iff N=R

(3) For any point p an element of R, N is a neighborhood for p in R_fc iff p is an element of N and there is a countable set A a subset of R st N = (R\A).

(4) For any point p an element of R and any e > 0, if N_e(p) = (p – e, p + e), then N_e(p) is a neighborhood for p in (R,T_R)

T_R = {A subset of R|a is an element of A implies (a – e,a + e) is a subset of A for some e >0)

(5) For any point p an element of R and any e > 0, if N_e(p) = [p, p + e), then N_e(p) is a neighborhood for p in R_S

(6) For any point p an element of R and any e > 0, if N_e(p) = (p – e, inf), then N_e(p) is a neighborhood for p in R_U