Tikhonov plank

The Tichonow plank is a special topological space considered in the mathematical sub-area of topology , which, due to its unexpected properties, often serves as a counterexample. This room is named after the Russian mathematician AN Tikhonov , who constructed it in 1930. Because of the transcription used in French, the name Tychonoff-Planke can also be found . Ordinal numbers are used for its construction .

definition

Let and be the smallest infinite or uncountable ordinal number. Furthermore, let and be the sets of all ordinal numbers from 0 to or , provided with the order topology . The Tikhonov plank is then the room ${\displaystyle \omega }$ ${\displaystyle \omega _{1}}$ ${\displaystyle [0,\omega ]}$ ${\displaystyle [0,\omega _{1}]}$ ${\displaystyle \omega }$ ${\displaystyle \omega _{1}}$

${\displaystyle T:=([0,\omega _{1}]\times [0,\omega ])\setminus \{(\omega _{1},\omega )\}}$

provided with the subspace topology of the product topology .

properties

• As a sub-room of the compact Hausdorff room, the Tichonow plank is a completely regular room .
• The Tikhonov plank is a locally compact space , as it is created by removing a point from a compact space.
• One can show that is not normal ; the two disjoint , closed sets and cannot be separated by open sets. is therefore an example of a completely regular but not normal room. ${\displaystyle T}$ ${\displaystyle \{(\omega _{1},n);\,n\in \omega \}}$ ${\displaystyle \{(\alpha ,\omega );\,\alpha \in \omega _{1})\}}$ ${\displaystyle T}$
• ${\displaystyle T}$is a subspace of the compact and therefore normal Hausdorff space . We therefore have an example of a non-normal open subspace of a normal room. Since all subspaces of completely normal rooms are normal again, is also an example of a not completely normal compact room. ${\displaystyle [0,\omega _{1}]\times [0,\omega ]}$ ${\displaystyle [0,\omega _{1}]\times [0,\omega ]}$
• ${\displaystyle [0,\omega _{1}]\times [0,\omega ]}$is not perfectly normal . One can show that there is no continuous function with and . This is because sets of zeros of continuous, real-valued functions are always sets, which is not the case with. ${\displaystyle f\colon [0,\omega _{1}]\times [0,\omega ]\rightarrow [0,1]}$ ${\displaystyle f^{-1}(\{1\})=\{(0,0)\}}$ ${\displaystyle f^{-1}(\{0\})=\{(\omega _{1},\omega )\}}$ ${\displaystyle G_{\delta }}$ ${\displaystyle \{(\omega _{1},\omega )\}}$

See also

• Dieudonné plank , a finer topology on the same basic set

literature

• Lynn Arthur Steen , J. Arthur Seebach: Counterexamples in Topology. 2nd edition. Springer, New York NY et al. 1978, ISBN 3-540-90312-7 , Example 86, (Reprinted by: Dover Publications, New York NY 1995, ISBN 0-486-68735-X ).
• Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. Vol. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 , section 4.5.
• Andrei Tychonoff : About the topological expansion of spaces. In: Mathematical Annals . Vol. 102, 1930, pp. 544-561, doi : 10.1007 / BF01782364 , digital version (PDF; 1.21 MB) .