# Definition:Strictly Precede/Definition 1

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## Definition

Let $\left({S, \prec}\right)$ be a strictly ordered set.

Let $a, b \in S$ and $a \prec b$.

Then **$a$ strictly precedes $b$**.

## Also known as

The statement **$a$ strictly precedes $b$** can be expressed as **$a$ is a strict predecessor of $b$**.

Some sources refer to a **strict predecessor** simply as a **predecessor**.

When the underlying set $S$ of the ordered set $\left({S, <}\right)$ is one of the sets of numbers $\N$, $\Z$, $\Q$, $\R$ or a subset, the term **is less than** is usually used instead of **(strictly) precedes**.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Orderings