Which space is homeomorphic to $X / \sim$

Question

In $\mathbb{R}^2$ defined the relation of equivalence $(a,x) \sim (a,y)$, for all $x,y \in \mathbb{R}$ if $a \neq 0$.I have to say which space is homeomorphic to $\mathbb{R}^2 / \sim$, can anybody give an idea?

Answer 1

HINT: Let $X=(\{0\}\times\Bbb R)\cup(\Bbb R\times\{0\})$, the union of the two coordinate axes in $\Bbb R^2$. There is a pretty obvious bijection from $\Bbb R^2/\sim$ to $X$. Try to define a topology on $X$ that makes that bijection a homeomorphism. It’s probably easiest to define basic open nbhds at each point of $X$.