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hornejada1c 2022-07-15 Answered
Let X = { ( x , y ) R 2 y = 0  or  x 0 } and f : X R defined by f ( x , y ) = x . I want to find C X closed such that f ( C ) R isn't closed. How to prove that f is a closed map?
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Kaylie Mcdonald
Answered 2022-07-16 Author has 19 answers
Step 1
f i understand correctly, X = { ( x , 0 ) : x < 0 } { ( x , y ) : x 0 } . So, if we take C = { ( x , 0 ) : x < 0 } { ( 0 , y ) : y R } then C is closed because X C is open in R 2 but when we apply f (which is the projection map actually, i.e. continuous 1-1 onto), we have f ( C ) = ( , 0 ) which is obviously open in R .

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