I am trying assignments in topology and I got stuck on this question:
Prove that the map $f:\Bbb R^2 \to \Bbb R$ defined by
$$f(x, y) = y^3 + xy^2 + x + y$$ is a quotient map.
I have done a course on topology but quotient maps were not covered in it. So, I am unable to do it.
I am using this definition of quotient map: A map $p$ is called a quotient map if $p: X\to Y$ is such that
(a) $p$ is surjective,
(b) $p$ is continuous,
(c) $U$ belonging to $Y$, $p^{-1}(U)$ open in $X$ implies $U$ is open in $Y$.
For my function, the first 2 conditions are satisfied but I am not able to prove that it satisfies the 3rd condition.
Kindly tell.
Use the final topology on $\mathbb{R}$.
Let $\mathcal{T}_{\mathbb{R}\times\mathbb{R}} = \mathcal{T}_1$ denote the standard topology on $\mathbb{R}^2$. Now consider the final topology on $\mathbb{R}$ induced by $f$, i.e. $$\mathcal{T}_2 = \{V \subset \mathbb{R} \mid f^{-1}(V) \in \mathcal{T}_{1}\}.$$
Now by construction, $f$ is obviously $\mathcal{T}_1-\mathcal{T}_2$ continous. Since you've already proven $f$ to be surjective it follows that $f$ is a quotient map.