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Is the given following statement istrue /false ?

Given $X = [-1,1] \times [-1,1]$ such that C ={$(x,y) \in X : xy= 0 $} and D ={$(x,y) \in X, x = {+}^{-} y$ }. Then $C$ is homeomorphic to $D$

First i was drawing the graphs

enter image description here

From the graph i noticed that it will not homomeorphics ...

is it true /false ?

Pliz help me...

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1 Answer 1

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A rotation of $\frac{\pi}{4}$ clockwise and then multiplying $x$ and $y$ by $\sqrt2$ respectively seems to be the homeomorphism you are searching for: bijective and bi-continuous.

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  • $\begingroup$ im not getting can u elaborate in details... $\endgroup$
    – jasmine
    May 27, 2018 at 21:04
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    $\begingroup$ To be specific, we map every $(x,y)\in C$ to $(x-y,x+y)$ and verify that $(x-y,x+y)$ is indeed in $D$. In fact, for every $(x,y)\in C$, it's of the form $(x,0)$ or $(y,0)$, so finally its image will turn out to be either $(x,x)$ or $(-y,y)$, which are in $D$. $\endgroup$
    – Muduri
    May 27, 2018 at 21:10
  • $\begingroup$ thanks @Maduri_ $\endgroup$
    – jasmine
    May 27, 2018 at 21:16

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