Prove Closed Neighborhoods in R's Topologies

(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

Re: Prove Closed Neighborhoods in R's Topologies

Was this question to determine **whether or not** these are true? Most of them are not true.

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Originally Posted by

**Kiefer** (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)

So you are asked to show that every singleton set, {p} is both open and close in the discrete topology.

By definition of "discrete topology", every singleton set is open. Because **all** unions of open sets are open, and **every** set is is a union of singleton sets, every set is open. Now, what is the definition of "closed set"?

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

Assuming you are still talking about the discrete topology, this is NOT true!

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

So this is a new topology. The condition that "a is an element of A implies R is a subset of A" (and, of course, "R is a subset of A" and "A is a subset of R" means that A= R) says that the only members of the "indscrete topology" are R itself and the empty set.

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(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)

Again, this is NOT true. A "closed neighborhood of p" is a closed set that contains p. And you just said that the only closed sets are R and the empty set.

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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)

So if a is in an open set, there exist some (perhaps very small) interval, (a- e, a+e), that is a subset of A.

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(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

Again, NOT true.

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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