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Non-Euclidean topology on the plane

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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.

Disks generate intervals.jpg

However, there are non-Euclidean topologies on the plane that do that.

(1) Vertical strips:

Vertical strips.jpg

VS < ET

(2) Vertical half-strips:

Vertical half-strips.jpg

VS < ET

(3) Horizontal intervals:

Horizontal intervals.jpg

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.