# Definition:Sorgenfrey Line

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## Definition

Let $\R$ be the set of real numbers

Let $\mathcal B$ be the set:

- $\mathcal B = \left\{{\left[{a \,.\,.\, b}\right): a, b \in \R}\right\}$

where $\left[{a \,.\,.\, b}\right)$ is the half-open interval $\left\{{x \in \R: a \le x < b}\right\}$.

Then $\mathcal B$ is the basis for a topology $\tau$ on $\R$.

The topological space $T = \left({\R, \tau}\right)$ is referred to as the **Sorgenfrey line**.

## Also known as

The **Sorgenfrey line** is also found in the literature referred to as:

- the
**lower limit topology** - the
**right half-open interval topology**.

## Also see

- Results about
**the Sorgenfrey line**can be found here.

## Source of Name

This entry was named for Robert Henry Sorgenfrey.

## Sources

- 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*: $\text{II}: \ 51$