In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Technically, Hopf found a many-to-one continuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct point of the 2-sphere comes from a distinct circle of the 3-sphere (Hopf 1931). Thus the 3-sphere is composed of fibers, where each fiber is a circle—one for each...
0
consider the three subsets of $\mathbb{R}^2$ namely
$$
\begin{split}
A_1 &= \left\{(x,y)\left| x^2 +y^2 \le 1 \right. \right\}\\
A_2 &= \left\{(1,y) | y \in \mathbb{R}\right\}\\
A_3 &= \{(0,2)\}
\end{split}
$$
then there always exist a continuous real valued function $f$ on $\mathbb{R}^2$ such t...
0
Let $A$ be a commutative ring where every element has a multiplicative norm $N$.
So if $a,b,c$ are in $A$, such that $ab = c$ Then $ N(a) N(b) = N(c)$ ( So $N$ is a multiplicative homomorphism )
Let $B$ be a commutative ring.
Usually we take $N$ to be integers.
But When $N$ is defined as ele...
We define the complex exponential function:
$$\begin{array}{rcl}
\exp:\Bbb C &\to& \Bbb C^\times \\
z &\mapsto& \exp(z)=\displaystyle{\sum_{n=0}^\infty \frac{z^n}{n!}.}
\end{array}$$
I wan't to show that this map is surjective. My idea is to show that the real exponential $\exp|_\Bbb R$ maps...
