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Section Relations

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, “\(\lt\)”, “\(=\)”, “\(\geq\)” on the integers, for example.

Definition 14.11.

A relation on a set \(S\) is a subset \(R\) of \(S \times S\text{.}\)

For example, the subset \(R = \{(a,a) \mid a \in \Z \}\) of \(\Z \times \Z\) 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 \((a,b) \in R = \{(a,a) \mid a \in \Z \}\text{.}\)

When discussing relations, there are three specific properties that we consider.

A relation \(\sim\) on a set \(S\) is reflexive if \(a \sim a\) for all \(a \in S\text{.}\)
A relation \(\sim\) on a set \(S\) is symmetric if whenever \(a \sim b\) in \(S\) we also have \(b \sim a\text{.}\)
A relation \(\sim\) on a set \(S\) is transitive if whenever \(a \sim b\) and \(b \sim c\) in \(S\) we also have \(a \sim c\text{.}\)

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

Definition 14.12.

An equivalence relation is a relation on a set that is reflexive, symmetric, and transitive.

Activity 14.5.

(a)

Explain why metric equivalence is an equivalence relation.

(b)

Explain why topological equivalence is 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.

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