Suppose we have a topological space $X = \mathbb{R}^2$ with standard topology
and an equivalence relation on $X$ defined as:
$(a,b) \sim (x,y) \iff a + b^2 = x + y^2$
Show that the quotient space is homeomorphic to topological space $Y$= $\mathbb{R}$ with standard topology (i.e there exists some function $f: X/\sim\to Y$ such that $f$ is continuous, bijective, and the inverse is continuous.
So I know f([(x,y)]) = x + y$^2$, works
Note that f is well defined since any (a,b) $\in$ [(x,y)], a + b$^2$ = x + y$^2$
How do I show that f is continuous and so is its inverse?
Here are some definitions/ theorems I know:
f is continuous iff for every open interval in Y, the preimage of the open interval is open in X\ $\sim$
The topology on X\ $\sim$ is { U $\subseteq$ X\ $\sim$ | p$^{-1}$(U) is open in X= $\mathbb{R}^2$} where p is the quotient map. This is known as the quotient topology.
Best Answer
HINT: Fix $c\in\Bbb R$, and look at the set of points $\langle x,y\rangle\in\Bbb R^2$ such that $x+y^2=c$: that’s one equivalence class. You can even think of it pictorially: it’s the graph of $x=c-y^2$.