Re: [分析] 高微(4)
※ 引述《PttFund (批踢踢基金)》之銘言：
: A metric space M is said to be locally path-connected if
: each point in M has a neighborhood U is path-connected.
: Show that (M is connected and locally path-connected)
: iff (M is path-connected).
Note: 在高微課中, path-connected ===> connected 是已經知道的事實,
而我們也曉得 connected =X=> path-connected, 反例是 cl(S),
where S = { x╳sin(1/x) | 0 < x≦1 }.
Sketck of proof:
1. Define a relation on M by saying x~y iff there is a path
from x to y. Show that this is an equivalent relation.
The equivalence classes are called the path components of M.
Clearly, M is path-connected iff M has exactly one path
component.
2. Show that M is locally path-connected => each path component
of M is open.
3. Deduce that a connected and locally path-connected space
is connected.
先提示到這邊.
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※ 發信站: 批踢踢實業坊(ptt.cc)
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