## Flying on the surface!

Remember the time when people thought that earth was flat? Nah, you probably don’t! Earth, or an approximated sphere/ellipsoid/ball sits inside $\mathbb{R}^3$. However, to local residents of our puny planet, it might look like a plane. The above math-fiction-mumbo-jumbo I mentioned begs me to define something rigorous.

Definition 1: Let $X$ and $Y$ be topological spaces. If there is a map $f$ from $X$ to $Y$ such that $f$ is a bijective, continuous and open map, then we say that such a $f$ is a homeomorphism and $X$ and $Y$ are said to be homeomorphic to each other.

Definition 2: Let $X$ and $Y$ be topological spaces. If for all $x \in X$, there is an open neighbourhood $U$ of $x$ such that $f(U)$ is open and $U$ is homeomorphic to $f(U)$, then $f$ is called a local homeomorphism.

Now, if we take our usual sphere, its surface is locally homeomorphic to $\mathbb{R}^2$. This is equivalent to saying that to a person on a large sphere(or our planet), the surface will look like a plane. To see the local homeomorphism, choose any point, take a small open disc on the surface around it. That will look like a lense. Flattening it out will give you a planar disc(Note that the map is injective, surjective onto the image, continuous and inverse continuous).

Now what would have happened if Earth was sitting in $\mathbb{R}^4$ and with an extra coordinate. To be precise, let us say the equation is

$x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1$.

Let us choose a point $y = (y_1 , y_2 , y_3 , y_4)$ on the surface of our new sphere. Take a small disc $D$. of radius $r$ centered at $y$. Now, if we “flatten” $D$ in a similar way, what we get is a solid sphere or a 3 dimensional disc. This means that our new sphere is locally homeomorphic to $\mathbb{R}^3$.

Therefore, walking on the surface of a 3-dimensional(as a manifold) sphere is the same as flying in $\mathbb{R}^3$.