推 sbshank :謝謝你!我在了解一下 10/23 16:19
※ 引述《sbshank (季)》之銘言：
: http://ppt.cc/B0xR
: 第一題的想法大概是實數上的開區間至多是可數的
: 但第二題完全沒有頭緒，似乎要用第一題的結果
: 謝謝!
提示：
16:
For each y in S, choose δ ＞ 0 such that (y, y + δ ) ∩ S
y y
is empty. S clearly has the same cardinality as
{(y, y + δ ) : y in S}
y
which is a family of disjoint open intervals, and any
family of disjoint open intervals is at most countable.
(The latter is clear because you can choose a distinct rational number
in each of the open intervals.)
17 (a):
Consider f:[0, 1]→Ｒ,
╭ 0 if 0 ≦ x ≦ 1/2,
f(x) = ┤
╰ 1 if 1/2 ＜ x ≦ 1.
17 (b):
Notation:
f([0, 1]) = {f(x) : x in [0, 1]}
is the image of f.
Let B be a countable base for Ｒ.
n
Say y = f(x) for some x in [0, 1]. There exists
δ > 0 such that y is the minimum of f on (x - δ, x + δ).
Choose n such that x is in B and B is contained in (x - δ, x + δ),
n n
and let
g(y) = n.
Then g : f([0, 1])→Ｎ is one-to-one, so f([0, 1]) is
at most countable.
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◆ From: 75.62.128.123
※ 編輯: cgkm 來自: 75.62.128.123 (10/20 13:44)
※ 編輯: cgkm 來自: 75.62.128.123 (10/20 13:49)
※ 編輯: cgkm 來自: 75.62.128.123 (10/20 14:50)