Tychonoff plank

The Tikhonov plank is, a relevant part in the mathematical field of topology specific topological space that is often used as a counter-example because of its unexpected properties. This room is named after the Russian mathematician AN Tikhonov, who designed it in 1930. Because of the transcription used in French as you can also find the name of Tychonoff plank. For its construction ordinals are used.

Definition

Let and the smallest infinite or uncountable ordinal. Next were and the amounts of all Ordninalzahlen from 0 to or provided with the order topology. The Tikhonov plank is then the space

Equipped with the subspace topology of the product topology.

Properties

  • The Tikhonov plank is a completely regular space as a subspace of a compact Hausdorff space.
  • The Tikhonov plank is a locally compact space, since it is generated by removal of a point from a compact space.
  • It can be shown that it is not normal; the two disjoint closed sets and can not be separated by open sets. is thus an example of a completely regular but not normal room.
  • Is the subspace of the compact and hence normal Hausdorff space. We have therefore an example of a non-normal open subspace of a normal space. Since all the sub-spaces completely normal areas are normal, is also an example of a not fully normal compact space.
  • Is not perfectly normal. It can be shown that there is not a continuous function and with. The reason is that zero sets of continuous, real-valued functions are always quantities, but this is not the case on.
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