Re: [分析] 高微(4)

[PttFund (批踢踢基金)]

※ 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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◆ From: 140.112.218.142
推 iddee:這題哪一本高微書上的阿 好陌生的題目 證明也不短
推 PttFund:不曉得 XD