Metrics

selinunan · #1 $Old$ 11-29-2008, 12:48 PM

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.

Plato · #2 $Old$ 11-29-2008, 01:38 PM

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