Definition:Half-Disc Topology
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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
- 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