Hopf map is continuous

Question

Consider map $h:\mathbb{S}^{3}\to \mathbb{S}^{2}$ defined as $h(a,b)=(a\bar{b}+b\bar{a},ib\bar{a}-ia\bar{b},|a|^{2}-|b|^{2})$

Does it just follow by seeing h as map from $\mathbb{C}^{4}$ to $\mathbb{R}^{3}$ or am I missing sth, and it follows from somehow showing it's preimage of open is open.

Answer 1

Let $f \colon X \to Y$ be a continuous mapping of topological spaces. Then given any subset $A$ of $X$, the restriction $g \colon A \to Y$ is continuous.

Furthermore, given any continuous mapping $g \colon A \to Y$, if $B$ is a subset of $Y$ such that $g[A] \subset B$, then the induced mapping $h \colon A \to B$ is continuous.

In the present situation, $X = \mathbb{C}^2$, $Y = \mathbb{R}^3$, $A = S^3$, $B = S^2$.

Answer 2

The proof of continuity of this map is the same as the proof of continuity of a function like $f(x)=\frac{x^2-3x}{5x+7}$, namely, the map can be expressed as a composition of very simple functions whose continuity is well known from any advanced calculus course.

For that rational function that I wrote, the simple functions involved are: squaring $x \to x^2$; scalar multiplication $x \to mx$ for any constant $m$; subtraction $(x,y) \to (x-y)$; addition $(x,y) \to x+y$; division $(x,y) \to x/y$ for $y \ne 0$; and a few others. All of those individual functions can be proved to be continuous by simple $\epsilon-\delta$ arguments that you learn in advanced calculus. Then, of course, you also prove and use the theorem that the composition of continuous functions is continuous.

For continuity of the Hopf map using the formula you gave, the simple functions involved are some of the above ones with real numbers plus some with complex numbers such as: complex conjugate $z \mapsto \bar z$; complex addition $(w,z) \to (w+z)$; complex subtraction $(w,z) \to w-z$; complex multiplication $(w,z) \to wz$; complex absolute value $z \mapsto |z|$.