# Definition:Chain (Set Theory)

## Contents

## Definition

Let $\left({S, \preceq}\right)$ be an ordered set.

A **chain in $S$** is a totally ordered subset of $S$.

Thus a totally ordered set is itself a **chain** in its own right.

### Length

Let $T$ be a chain in $S$.

Let $T$ be finite and non-empty.

The **length** of the chain $T$ is its cardinality minus $1$.

## Also defined as

Some sources use the term **chain** to mean the same thing as totally ordered set. While this is perfectly valid, as there is no source of confusion here, such usage is surprisingly uncommon.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Orderings - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 16$: Zorn's Lemma - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$