To formalize path homotopy as a way of distinguishing certain topological spaces, we need to introduce the notion of an *equivalence relation* and an *equivalence class*. We'll introduce the former here.

Consider a set *A* of objects. We pick certain pairs of elements in *A* and say they have a particular *relation* to one another. In other words, a **relation** R on *A* (or more accurately, a **binary relation** on *A*) is simply a choice of pairs—a subset of *A* x *A*. If <a, b> is in R then we say that *a* has the relation R with *b*. We can also write this *a*R*b*.

If *A* is a set of people, R might be something like "is the father of". And so if *d* is Darth Vader and *l* is Luke Skywalker, then *d*R*l*.

A relation is said to be an **equivalence relation** iff it is a relation with the following properties:

- Reflexivity: xRx for all x in A
- Symmetry: if xRy then yRx
- Transitivity: if xRy and yRz then xRz

Our "is the father of" relation violates all three and so it certainly not an equivalence relation.

Something like "is less than" on the set of reals is transitive but not reflexive or symmetric and so is not an equivalence relation.

Something like "is less than or equal to" on the set of reals is transitive and reflexive but still not symmetric and so is not an equivalence relation.

Equality is an equivalence relation as it has all three necessary properties. Two topological spaces being homeomorphic is also an equivalence relation.

Importantly for us, *path homotopy* is an equivalence relation.

**UPDATE**: next post

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The original post was in the category: poincare_project but I'm still in the process of migrating categories over.