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posted by  Michael25 on 11/3/2009 3:24:59 PM  |  status: Closed  |  Earned Karma: 156

Analysis: Continuous?

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N/A N/A N/A N/A 11/5/2009 at 12:00:00 PM
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If u is open  f(u) is open, then f is called an open map. An open map need not be continuous. Suppose however, that f : X Y is one-to-one, onto, and open.
Show that f -1 is continuous.
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Oracle
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posted by tdv (MNV) on 11/5/2009 7:10:28 AM  |  status: Live
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Response Details:
Since  f is one to one and onto, f -1 :Y ---> X  is a well defined map.
Also we know that a map g : A---> B is continuous
                 if and only if g-1 (V) is open for evey openset V in B.-------(*)

Let V  be any open set  in X. Then 
Now (f-1)-1(V) = f(V) a open set by the assumption.
  This proves  f-1 is a continuous function (by (*).)
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