Relation as the Union of 4 Relations

Question

I'm trying to write the relation $$\rho=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: |x|+2|y|=1\}$$ as the union of 4 relations. Is it enough to just think of this as a diamond and use the four sides as 4 different relations? What I mean is $$\alpha=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: x+2y=1, 0\lt{x}\le{1},0\le{y}\lt{1/2}\}$$ $$\beta=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: x-2y=1, 0\le{x}\lt{1},-1/2\le{y}\lt{0}\}$$ $$\gamma=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: -x+2y=1, -1\lt{x}\le{0},0\lt{y}\le{1/2}\}$$ $$\delta=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: -x-2y=1, -1\le{x}\lt{0},-1/2\le{y}\lt{0}\}$$ I've mapped it out and I think this is exactly what is asked of me since now $$\alpha\cup\beta\cup\gamma\cup\delta=\rho$$ This last question was homework but the next is just me being inquisitive. How do you deform the above relation $\rho$ to make a circle? What is the continuous function? I know since these are both connected and continuous there are ways in topology to do this but I've not taken topology (doing set theory first) and was just curious.

Answer 1

You’ve written $\rho$ correctly as the union of four (pairwise disjoin) relations on $\Bbb R$; if there are no other conditions to be met, then you’ve done what was required.

Perhaps the easiest way to deform the set $\rho$ to a circle is to deform it to the unit circle $S^1=\{\langle x,y\rangle\in\Bbb R\times\Bbb R:x^2+y^2=1\}$ by expanding it radially. For each $p=\langle x,y\rangle\in\rho$ let $r_p=\sqrt{x^2+y^2}$, the distance from $p$ to the origin, and let $e_p=\frac1{r_p}$; then $\frac12\le r_p\le 1$, so $1\le e_p\le 2$. For each $p\in\rho$, $e_pp\in S^1$: $e_p$ is the factor be which $p$ must be expanded to move it out to $S^1$.

Now define

$$f:\rho\times[0,1]\to\Bbb R\times\Bbb R:\langle p,t\rangle\mapsto\big(1+t(e_p-1)\big)p\;;$$

then $f$ is continuous, $f(p,0)=p$ for each $p\in\rho$, and $f(p,1)=e_pp\in S^1$ for each $p\in\rho$. That is, if $X_t=\{f(p,t):p\in\rho\}$ for each $t\in[0,1]$, then as $t$ moves from $0$ to $1$, $\rho=X_0$ is deformed continuously through the sets $X_t$ with $0<t<1$ to $X_1=S^1$.