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Math Help - Metrics

  1. #1
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    Metrics

    Let M be some nonempty set

    a) Let d be a metric on M and P ∈ M; Q ∈ B (P, r) . Show that
    B (Q, s) ⊂ B (P, r + s)

    b) Let d be a metric on M and P,Q ∈ M, T ∈ B (P, r) ∩ B (Q, r) . Find
    some s > 0 so that B (T, s) ⊂ B (P, r) ∩ B (Q, r)

    c) Suppose that d₁ and d₂ are two metrics on the set M such that for
    all x, y ∈ M d₁ (x, y) ≤ 2d₂ (x, y) . Denote by B₁ (p, r) and B₂ (p, r)
    the open balls with center p and radius r with in the metrics d₁ and
    d₂ respectively. Show that B₂ (p, r) ⊂ B₁ (p, 2r)


    If you help me, I will be really happy.
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  2. #2
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    Quote Originally Posted by selinunan View Post
    Let M be some nonempty set
    a) Let d be a metric on M and P ∈ M; Q ∈ B (P, r) . Show that
    B (Q, s) ⊂ B (P, r + s)

    b) Let d be a metric on M and P,Q ∈ M, T ∈ B (P, r) ∩ B (Q, r) . Find
    some s > 0 so that B (T, s) ⊂ B (P, r) ∩ B (Q, r)
    a) Hint: If X \in B(Q;s) then d(P,X) \leq d(P,Q)+d(Q,X)

    b) Hint: s=\frac {\min\{r-d(P,T),r-d(Q,T)\}}{2}
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