# Is the given following statement is true /false relating to homeomorphic?

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

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

is it true /false ?

Pliz help me...

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.

• im not getting can u elaborate in details... May 27, 2018 at 21:04
• 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 $D$. May 27, 2018 at 21:10
• thanks @Maduri_ May 27, 2018 at 21:16