Definition:Irregular Lattice Topology
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Definition
Let $S$ be the subset of the lattice points of the Cartesian plane defined as:
- $S := \set {\tuple {i, j}: i, j \in \Z_{>0} } \cup \set {\tuple {i, 0}: i \in \Z_{\ge 0} }$
Let each point of $S$ be defined as being open.
Let each point of $S$ of the form $\tuple {i, 0}$ such that $i \ne 0$ have as a local basis sets $U_n$ of the form:
- $\map {U_n} {i, 0} := \set {\tuple {i, k}: k = 0 \text { or } k \ge n}$
Let the point $\tuple {0, 0}$ have as a local basis a set $V_n$ of the form:
- $V_n := \set {\tuple {i, k}: i = k = 0 \text { or } i, k \ge n}$
Let $\tau$ be the topology generated from all these $U_n$ and $V_n$.
$\tau$ is referred to as the irregular lattice topology.
Also see
- Results about the irregular lattice topology can be found here.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.): Part $\text {II}$: Counterexamples: $79$. Irregular Lattice Topology