Definition:Sorgenfrey's Half-Open Square Topology
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Definition
Let $A = \struct {\R, \sigma}$ be a Sorgenfrey line.
Let $S = A \times A$ denote the product space of $A$ with itself, whose topology $\tau$ is such that:
- the neighborhood of a point $\tuple {p, q}$ is a rectangle in $\R^2$ of the form:
- $\set {\set {x, y}: p \le x < p + \epsilon_1, q \le y < q + \epsilon_2}$
- for some $\epsilon_1, \epsilon_2 \in \R_{>0}$.
$\struct {S, \tau}$ is referred to as Sorgenfrey's half-open square topology.
Also see
- Results about Sorgenfrey's half-open square topology can be found here.
Source of Name
This entry was named for Robert Henry Sorgenfrey.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $84$. Sorgenfrey's Half-Open Square Topology