With continuity under our belts (albeit with some bumps along the way), we can look at something that many people consider *the* central concept of topology: homeomorphisms.
A homeomorphism is what defines the topological concept of *equivalence*. Remember the clay mug/torus metaphor from from my introduction: in topology, two topological spaces are equivalent if they can be bent, stretched, smushed, twisted, or glued to form the same shape *without* tearing.
The rest is beneath the fold.
*(A very important thing about the intuition above: understanding homeomorphism in terms of deforming topological spaces, isn’t really quite correct. The reshaping of topological spaces is more properly modeled as a continuous deformation, which we’ll define later in terms of something called **isotopy**.)*
To make it more general, so that we aren’t limited to topological spaces that are easily visualized, we can say that two topological spaces are equivalent if they can be reshaped into each other without making any points that are close together in one be separated in the other.
That’s a fairly awkward way of saying it. It’s actually easier to talk about it
in formal terms – which is where homeomorphisms come into play.
A homeomorphism between two topological spaces **S** and **T** is a function f : **S** → **T**, such that:
1. f is continuous
2. f-1 is continuous
3. f is a bijection (f is total, one-to-one, and onto).
By the continuity of f, we know that the mapping from **S** to **T** is smooth; by the continuity of its inverse, we know that the mapping from **T** to **S** is smooth; and since it’s a bijection, we know that every point in **S** is mapped to exactly one point in **T**, and every point in **T** is mapped to by exactly one point in **S**.
A homeomorphism is also known as a *topological isomorphism*: it’s an isomorphism between two topological spaces – that is, a function between two topological spaces that preserves all of their topological properties. Every homeomorphism is also an iso-arrow in the category of topologies: everything we know about iso-arrows in categories can be specialized to apply to the relationships between topological spaces defined by homeomorphisms.
There are some things about homeomorphisms that can seem a little counterintuitive. Homeomorphisms determine a kind of topological equivalence: they preserve all topological properties. But we often associate properties with topological spaces that are *not* really topological. We often think of topological spaces in terms of metric spaces that have topological properties. But *metric* properties are *not* topological properties, and so a homeomorphism can modify the metric properties of a space.
Homeomorphisms are fascinating things. They give you an ability to grasp what topological properties really mean in a fascinating way. A few examples:
* A sphere and a cube are the topologically identical. Not too surprising, but it sometimes throws people that a hard angles on a cube can turn into a continuous curves of a sphere.
* A more interesting one: you can “shrink” the infinite to a finite range! For any number *n*, the open numeric interval from -n to n is homeomorphic to the set of all real numbers.
* And finally, a really fascinating one that *I* at least was totally surprised by when I first learned it: take a sphere. Remove *one* point. What you get is homeomorphic to a plane. (I just find this one too cool for words: in topology, a sphere is only different from a plane because of *one* point!)