又係metric

kwoklap

Suppose d and d' are two metrics on X satisfying d'(x, y) <= c d(x, y) for some constant c > 0. Prove that d'-open subsets are d-open.

Ans:  
The relation d' ≤ c d implies that Bd(x, c^(-1)ε) ⊂ Bd'(x, ε). Then
U ⊂ X is d'-open
⇒ For any x ∈ U, there is ε > 0, such that Bd'(x, ε) ⊂ U
⇒ For any x ∈ U, there is ε > 0, such that Bd(x, c^(-1)ε) ⊂ U
⇒ U ⊂ X is d-open

我睇唔明紅色果句,點解個metric大左個ball會細左

Naozumi

Let y be an element in Bd(x, c^(-1)ε)
i.e. d(x, y) < ε/c
d'(x, y) < = c d(x,y) < ε
so y is also an element in Bd'(x,ε)

kwoklap

Thank you very much
今個sem玩命讀topology

kwoklap

其實我想問下做metric continuity的題目有冇方法係大約估到佢係continuous or discontinuous

如果知道佢continuous, 可以從定義的方向去做
or 知道佢係discontinuous,可以從counter example的方向去做

次次做題目都要諗一大餐,好似下面果條
Determine the continuity for the L1- and L∞-metrics
M(f) = max[0,1] f : C[0,1] → R usual

到底應該從proof continuous定係disproof continuous出發
其實呢條我做左,只係想問下你做果時點樣入手(諗法)

[ 本帖最後由 kwoklap 於 11-2-20 08:26 PM 編輯 ]