Definition:Half-Disc Topology

Definition

Let $P = \set {\tuple {x, y}: x \in \R, y \in \R_{>0} }$ be the open upper half-plane.

Let $\struct {P, \tau_d}$ be the open upper half-plane with the Euclidean topology.

Let $L$ denote the $x$-axis

Let $\BB$ be the set of sets of the form:

$\set x \cup \paren {U \cap P}$

where:

$x \in L$

$U$ is a Euclidean neighborhood of $x$.

Let $\tau^*$ be the topology generated from $\BB$.

$\tau^*$ is referred to as the half-disc topology.

Also see

• Half-Disc Topology is Topology

• Results about the half-disc topology can be found here.

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $78$. Half-Disc Topology