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Show that both [itex]Y\cup A[/itex] and [itex]Y\cup B[/itex]

  1. Aug 21, 2014 #1
    1. The problem statement, all variables and given/known data

    Suppose [itex]Y\subset X[/itex] and [itex]X,Y[/itex] are connected and [itex]A,B[/itex] form separation for [itex]X-Y[/itex] then, Prove that [itex]Y\cup A[/itex] and [itex]Y\cup B[/itex] are connected.




    3. The attempt at a solution

    I can show easily that at least one of [itex]Y\cup A[/itex] or [itex]Y\cup B[/itex] is connected. but I don't know how to show the required

    Any hints?

    Here are some facts that I know about connectdness that may be helpful.

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    As a way to proceed, we can suppose that There is a seperation [itex]C \cup D[/itex] of,say, [itex]Y\cup A[/itex] So it's clear that [itex]Y [/itex] included in either [itex]C[/itex] or [itex]D[/itex]
     
    Last edited by a moderator: Aug 21, 2014
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  3. Aug 22, 2014 #2

    haruspex

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    I'd say that's good start, and you can choose C to be the one containing Y.
    Maybe you can now show that D, BUC form a separation of X? Or something like that.
     
  4. Aug 26, 2014 #3

    andrewkirk

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    Yes that is how I would start. Let's call C the part that includes Y.
    Next we can use the lemma (1.1 on page 47 of my very old edition of 'Topology - a first course' by Munkres) that if two sets form a separation then neither contains any limit points of the other, So C contains no limit points of D, and also B contains no limit points of D because ##D\subset A## and A,B is a separation of X-Y.

    But then ##D, C\cup B## is a separation of ##X## because
    ##\{x\vert x## is a limit point of ##C\cup B\}\subset \{x\vert x## is a limit point of ##C\}\cup\{x\vert x## is a limit point of ##B\}\subset X-D##

    Since ##X## is connected we conclude that there can be no separation ##C,D##.
     
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