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Non-Euclidean topology on the plane
For the background, see Neighborhoods and topologies.
- The standard basis of the Euclidean topology on the plane consists of open disks.
- The standard basis of the Euclidean topology on the line consists of open intervals.
Via relative topology, the former topology on X = the plane generates the latter on A = the x-axis.
However, there are non-Euclidean topologies on the plane that do that.
(1) Vertical strips:
VS < ET
(2) Vertical half-strips:
VS < ET
(3) Horizontal intervals:
HI > ET
Exercise. Prove that (a) the above collection form basis topologies, (b) these topologies are non-euclidean, (c) they generate the Euclidean topology on the horizontal lines, (d) they generate the indicated topologies on the vertical lines.
Exercise. Find a topology that generates the Euclidean topology on both horizontal and vertical lines.
Note: Minkowski spacetime is non-Euclidean.