Definition:Double Origin Topology

Definition

 * Double-origin-topology.png

Let $\struct {\R^2, \tau_d}$ be the real number plane with the usual (Euclidean) topology.

Let $0^*$ be an additional point.

For all $z \in \R^2$, let a neighborhood basis $\BB$ be constructed as follows.

For $z \in \R^2: z \notin \set {\tuple {0, 0}, 0^*}$, the neighborhoods of $z$ are the usual open sets of $R^2 \setminus \set {\tuple {0, 0} }$.

For $z \in \tuple {0, 0}$ and $0^*$, let us take as a neighborhood basis the sets:

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

$\tau$ is referred to as the double origin topology.

Also see

 * Double Origin Topology is Topology