Show that some map is a quotient map

Question

Let $f: [0,1] \times [0,1]$ be $f(x_1, x_2) = (x_1, (1-x_1), (1-x_1)(1-x_2))$. Notice that this is a map from $[0,1] \times [0,1]$ onto $\{(y_1, y_2, y_3)| y_1+y_2+y_3=1, 0 \leq y_1 \leq 1, 0 \leq y_2 \leq 1, 0 \leq y_3 \leq 1\}$. It is obviously continuous. The domain is a compact set and the image is a Hausdorff space, thus $f$ must be closed. But how do I show that $f$ is a quotient map?

Answer 1

Outside of the problem identified by @AjayKumarNair that you wrote down the wrong image, you can use this theorem:

For every compact $X$ and every Hausdorff $Y$, every continuous surjective $f : X \to Y$ is a quotient map.

For the proof, suppose $A \subset Y$ and $f^{-1}(A)$ is open. It follows that $X - f^{-1}(A) = f^{-1}(Y-A)$ is closed. So $f^{-1}(Y-A)$ is compact (because $X$ is compact). So $Y-A = f(f^{-1}(Y-A))$ is compact. So $Y-A$ is closed (because $Y$ is Hausdorff). So $A$ is open.