Category:Double Origin Topology

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This category contains results about Double Origin Topology.

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:

\(\ds \map {V_n} {0, 0}\) \(=\) \(\ds \set {\tuple {x, y}: x^2 + y^2 < \dfrac 1 {n^2}, y > 0} \cup \set {\tuple {0, 0} }\)
\(\ds \map {V_n} {0^*}\) \(=\) \(\ds \set {\tuple {x, y}: x^2 + y^2 < \dfrac 1 {n^2}, y < 0} \cup \set {0^*}\)


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


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

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