※ 引述《Johnniekaka (卡卡的..)》之銘言：
: Let A1>A2>A3... be a decreasing sequence of
: connected compact subsets of R_n.
: Show that intersection among Ai (i>=1) is connected.
: 請教各位高手，給予指點，謝謝!
first note that A1, A2, A3,......... is nonempty, decreasing, closed and
bounded.
by Cantor intersection theorem
無窮大
=>交集 Ak is closed & nonempty, say T
k=1
let f be a 2-valued function on T
it suffices to prove that f is constant.
now suppose that f is not constant
=> 存在 x1, x2 屬於 T such that f(x1) =/= f(x2)
x1, x2 屬於 T => x1, x2 屬於 Ai, for all i = 1, 2, ...........
since Ai is connected for all i = 1, 2, ........... ,
f is constant on Ai for all i = 1, 2, ........... .
there is a contradiction.
thus, f is constant on T
(in this proof, we use the theorem : a metric space S is connected iff
every 2-valued function on S is constant.)
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◆ From: 140.119.96.6
※ 編輯: hanabiz 來自: 140.119.96.6 (08/21 15:09)