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Math Help - Proofs of neighborhood in R topologies

  1. #1
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    Proofs of neighborhood in R topologies

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

    (2) For any point p an element of R, N is a neighborhood for p in R_* iff N=R
    R_*={A is a subset of X|a is an element of A implies X is a subset of A} (indiscrete topology)

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

    (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
    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)

    (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
    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)
    Last edited by Kiefer; September 8th 2012 at 04:39 PM.
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  2. #2
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    Re: Proofs of neighborhood in R topologies

    Quote Originally Posted by Kiefer View Post
    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
    Sorry to tell you this, but your prof/textbook is using completely non-standard notation.
    So you must post the meaning of all symbols. What does each of R*, R_*, R_fc, & N_e(p) mean.
    Thanks from Kiefer
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  3. #3
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    Re: Proofs of neighborhood in R topologies

    Attempted proofs:
    (1) Claim 1: If N is a nbhd of p in R*, then p is an element of N
    Proof: Suppose N is a nbhd of p
    We know G is an element of R* such that p is an element of G and G is a subset of N (definition of nbhd)
    Hence p is an element of N
    Claim 2: If p is an element of N, then N is a nbhd of p in R*
    Proof: Suppose p is an element of N
    Claim 2.1: There exists G as an element of R* such that p is an element of G and G is a subset of N
    Proof: We know p is an element of N, N is a subset of N and N is an element of R*
    Therefore, there exists G in R* such that p is an element of G and G is a subset of N

    (2) Claim 1: If N is a nbhd of p in R_*, then N=R
    Proof: Suppose N is a nbhd of p in R_*
    Claim 1.1: N is a subset of R
    (((((PROVE Claim 1.1))))))
    Claim 1.2: R is a subset of N
    (((((PROVE Claim 1.2))))))
    Hence N =R
    Claim 2: If N=R, then N is a nbhd of p in R_*
    Proof: Suppose N=R
    Claim 2.1: There exists G as an element of R such that p is an element of G and G is a subset of N
    Claim 2.1.1: N is an element of R_*
    (((((PROVE Claim 2.1.1))))))
    We know p is an element of R
    So p is an element of N and N is a subset of N

    (3) Claim 1: If N is a nbhd of p in R_fc, then p is an element of N and there is a finite set A in R such that N=(R\A)
    Proof: Assume N is a nbhd of p in R_fc
    Claim 1.1: p is an element of N and there is a finite set A in R such that N=(R\A)
    Proof:We know F is an element of R_fc such that p is an element of G and G is a subset of N (def of nbhd)
    So p is an element of N
    ((((((Not sure about these lines))))
    ==>We know R_fc(A)={A is a subset of R|a is an element of A=>(R\A) is finite)}
    ==>Since p is an element of N, we know N is a subset of R and p is an element of N, so N=(R\A)
    Claim 2: If p is an element of N and there is a finite set A in R such that N=(R\A), then N is a nbhd
    (((((PROVE Claim 2))))))

    (4) Claim 1: (a,b) is an element of T_R for all a,b which are elements of R
    Proof:
    Claim 1.1: (a,b) is a subset of R
    Proof: have show in class
    Claim 1.2: If p is an element of (a,b), then there exists an e>0 st (p e, p+e) is a subset of (a,b)
    Proof: Suppose p is an element of (a,b)
    Take e=min{(p a),(b p)}
    Claim 1.2.1: e>0 and (p e, p+e) is a subset of (a,b)
    We know a<p<b and e=p a >0 or e=b p >0
    So e>0
    e=p a => or a=p e => a is less than or equal to p e
    e=b p >0 or p+e=b => b is greater than or equal to p + e
    Clearly, p e < p + e and (p e, p+e) is a subset of (a,b)

    I feel pretty good about (1) and (4) but wouldn't mind them double checked. I got stuck on the marked parts of (2) and (3) and could use some help on approaching (5) and (6)
    Last edited by Kiefer; September 8th 2012 at 07:28 PM.
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