(1) For any point p, where p is an element of R, {p} is an open-and-closed neighborhood of p in R*

R represents the real numbers

R*={A is a subset of R|a is an element of A implies {a} is a subset of A} (discrete topology)

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

R_*={A is a subset of R|a is an element of A implies R is a subset of A} (indiscrete topology)

(3) For any point p, where p is an element of R, and any e>0, [p-e, p+e] is a closed neighborhood of 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)

(4) For any point p, where p is an element of R, and any e>0, [p, p+e] is an open-and-closed neighborhood of p in R_S

R_S = {A is a subset of R | a is an element of A implies [a, a + e) is a subset of A for some e > 0} (Sorgenfrey topology)

(5) F is closed in R_fc iff F is finite or F=R

R_fc={A is a subset of R|a is an element of A implies (R\A) is finite} (finite-complement topology)

(6) N is a closed neighborhood of p in R_fc iff N=R

(7) A nonemepty set F is closed in R_U iff F=(-infinity,b] for some b, which is an element of R, or F=R

R_U = {A is a subset of R | a is an element of A implies (a - e, +inf) is a subset of A for some e > 0} (upper topology)

(8) N is a closed neighborhood of p in R_U iff N=R