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From Intelligent Perception

Homeomorphisms are transformations that preserve topological properties.

It is required that both the function $$f: X {\rightarrow} Y $$ and its inverse, below, $$f^{-1}: Y {\rightarrow} X$$ are continuous.

Thus, for example, gluing is not a homeomorphism: gluing itself is continuous but its inverse - tearing - is not (see Examples of maps).

However you are allowed to cut if you glue it back together exactly as before. For example this is how you can un-knot this knot:

Exercise. What is this, topologically?

Let's make this idea more precise. Recall a couple of definitions.

A function $f: X {\rightarrow} Y$ is called one-to-one, or injective, if $$f(x) = f(y) \Rightarrow x = y.$$ Or, the preimage of a point, if non-empty, is a point.

A function $f: X {\rightarrow} Y$ is called onto, or surjective, if

Or, the image of the domain space is the whole target space, $f(X) = Y$.

A function that is one-to-one and onto is also called bijective.

Theorem. A function $f:X {\rightarrow} Y$ is bijective then it has the inverse:

Definition. Suppose $X$ and $Y$ are topological spaces and $f:X {\rightarrow} Y$ is a function. Then, $f$ is called a homeomorphism if

• $f$ is bijective,
• $f$ is continuous,
• $f^{-1}$ is continuous.

Then $X$ and $Y$ are called homeomorphic.

Theorem. Closed intervals of non-zero, finite length are homeomorphic.

Proof. Let $X = [a,b], Y = [c,d], b>a, d>c$. We will find a function $f: [a,b] {\rightarrow} [c,d]$ with $f(a) = c$ and $f(b) = d$.

The simplest function of this kind known to be continuous is linear:

To find the formula, use the point-slope formula from calculus. The line passes through $(a,c)$ and $(b,d)$, so its slope is $m = \frac{d-c}{b-a}$. Hence, the line is given by $$f(x) = c + m(x-a).$$ We can also give the inverse explicitly: $$f^{-1}(y) = a + \frac{1}{m} \cdot (y-b).$$ Also, $[a,b]$ can't be homeomorphic to a point because this function can't be bijective. $\blacksquare$

Similarly...

Theorem. Open intervals of finite length are homeomorphic.

Proof. Exercise. $\blacksquare$

Theorem. An open interval is not homeomorphic to a closed interval (nor half-open).

Proof. See Compactness. $\blacksquare$

Theorem. All open intervals, even infinite ones, are homeomorphic.

Proof. Tangent gives you a homeomorphism between $(- \pi /2, \pi /2)$ and $(- \infty , \infty )$.

$\blacksquare$

Another way to justify this conclusion is given by the following construction:

Here the "north pole" $N$ is taken out from a circle to form $X$. Then $X$ is homeomorphic to a finite open interval, and to an infinite interval, $Y$. The function $f:X {\rightarrow} Y$ is defined as follows:

Exercise. Prove that this $f$ is a homeomorphism.

The above construction is a 2D version of the stereographic projection:

which is, literally, a map.

This construction is used to prove that the sphere with a pinched point is homeomorphic to the plane.

Example. Show that the sphere and the hollow cube are homeomorphic. The idea is that if you insert a balloon inside a box you can then inflate it and fill the box from the inside.

We can instead concentrate on the inverse of $f$. Let's illustrate the idea in dimension $2$, i.e., square $Y$ and circle $X$. One can give an explicit representation of the function: $$f^{-1}(u) = \frac{u}{||u||},$$ where $||u||$ is the norm of $u$. The norm is known to be continuous and, therefore, so is its restriction to $Y$. Next, the ratio of two continuous functions is continuous, as we know from calculus, as long as the denominator isn't $0$ (it isn't on $Y$). Hence $f^{-1}$ is continuous. This function is basically a radial projection (see Examples of maps).

Theorem. Homeomorphism creates an equivalence relation on the set of all topological spaces.

Then it makes sense to call two spaces topologically equivalent if they are homeomorphic. We use the following notation for that: $$X {\approx} Y.$$

Theorem. If two topological spaces are homeomorphic then their homologies are isomorphic.

In other words, homology is a topological invariant. However, the converse isn't true. Indeed, even though points and cells are not homeomorphic, their homology coincide.

Sometimes you don't need to study the inverse directly.

Theorem. Suppose $X$ is compact and $Y$ is Hausdorff. If $f: X {\rightarrow} Y$ is continuous, one-to-one, and onto, then $f$ is a homeomorphism.

Properties preserved under homeomorphisms are called topological invariants.