Topology: An Inquiry-Based Approach

We use the word “equivalent” deliberately when talking about metric or topological equivalence. Recall that equivalence is a word used with relations, and that a relation is a way to compare two elements from a set. We are familiar with many relations on sets, “$<$”, “$=$”, “$\ge$” on the integers, for example.

For example, the subset $R=\{(a,a)\mid a\in \mathbb{Z}\}$ of $\mathbb{Z}\times \mathbb{Z}$ is the relation we call equals. If $R$ is a relation on a set $S\text{,}$ we usually suppress the set notation and write $a\sim b$ if $(a,b)\in R$ and say that $a$ is related to $b\text{.}$ In this case we often refer to $\sim$ as the relation instead of the set $R\text{.}$ Sometimes we use familiar symbols for special relations. For example, we write $a=b$ if $(a,b)\in R=\{(a,a)\mid a\in \mathbb{Z}\}\text{.}$

When we use the word “equivalence”, we are referring to an equivalence relation.

Equivalence relations are important because an equivalence relation on a set $S$ partitions the set into a disjoint union of equivalence classes. Since topological equivalence is an equivalence relation, we can treat the spaces that are topologically equivalent to each other as being essentially the same space from a topological perspective. Under the relation of homeomorphism we call the equivalence classes homeomorphism classes.