Binormal space

Binormal space is a term from the mathematical branch of topology . The term has, among other things, meaning for homotopy investigations in finite-dimensional real coordinate space .

definition

A topological space is called binormal if it is a normal Hausdorff space and at the same time countable paracompact in the sense that every at most countable open cover has a locally finite refinement . ${\displaystyle X}$

example

According to Arthur Harold Stone's theorem, a metric space is always paracompact , consequently also countably paracompact and, moreover, always normal. Therefore every metric space is binormal.

Characterization set

The following characterization theorem applies, which essentially goes back to a work by the Canadian mathematician Clifford Hugh Dowker from 1951:

The following conditions are equivalent for a Hausdorff room: ${\displaystyle X}$

(1) is binormal. ${\displaystyle X}$

(2) If there is any compact metric space , the associated product space is always a normal space. ${\displaystyle Y}$ ${\displaystyle X\times Y}$

(3) There is at least one infinite compact metric space for which the associated product space is a normal space. ${\displaystyle Y}$ ${\displaystyle X\times Y}$

(4) The product space formed with the closed unit interval is a normal space. ${\displaystyle I=[0,1]\subset \mathbb {R} }$ ${\displaystyle X\times I}$

(5) The product space formed from and the Hilbert cube is a normal space. ${\displaystyle X}$ ${\displaystyle X\times {I^{\aleph _{0}}}}$

(6) For every two semi-continuous real-valued functions in such a way that it is above- semi-continuous and below- semi-continuous and that the inequality always applies, there is a continuous function which always satisfies the relationship . ${\displaystyle f,g\colon X\rightarrow \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f(x)<g(x)\;\;(x\in X)}$ ${\displaystyle h\colon X\rightarrow \mathbb {R} }$ ${\displaystyle f(x)<h(x)<g(x)\;\;(x\in X)}$

Homotopy continuation clause

In connection with the binormality property, Borsuk's homotopy continuation applies , which goes back to a work by the Polish mathematician Karol Borsuk from 1937. This can be formulated as follows:

A binormal space and a closed subspace as well as two continuous mappings from into the sphere are given . ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle f_{0},f_{1}\colon A\rightarrow S^{n}}$ ${\displaystyle A}$ ${\displaystyle n}$ ${\displaystyle S^{n}\;(n\in \mathbb {N} )}$

Let and be homotop and have a constant continuation . ${\displaystyle f_{0}}$ ${\displaystyle f_{1}}$ ${\displaystyle f_{0}}$ ${\displaystyle F_{0}\colon X\rightarrow S^{n}}$

Then:

Also has a steady continuation , which is also homotop . ${\displaystyle f_{1}}$ ${\displaystyle F_{1}\colon X\rightarrow S^{n}}$ ${\displaystyle F_{0}}$

However, in 1975 Kiiti Morita and Michael Starbird independently proved that this is still valid even if the binormality property is left aside, i.e. if one only assumes that it is a normal Hausdorff space. ${\displaystyle X}$

Corollary

The Borsuk homotopy continuation theorem, in connection with the fact that the real coordinate space is a contractible space , leads to the following interesting corollary:

Let be a closed subset of the real coordinate space and be a continuous mapping. ${\displaystyle A}$ ${\displaystyle \mathbb {R} ^{k}\;(k\in \mathbb {N} )}$ ${\displaystyle f\colon A\rightarrow S^{n}}$

Then:

${\displaystyle f}$is nullhomotop if and only if has a continuous continuation . ${\displaystyle f}$ ${\displaystyle F\colon \mathbb {R} ^{k}\rightarrow S^{n}}$

Dowker's problem

Clifford Hugh Dowker raised the following question in his 1951 paper:

• Is a normal Hausdorff room always a countable paracompact room?

In other words:

• Is every normal Hausdorff space already binormal?

The photo associated with this issue problem is called dowkersches problem ( English Dowker's problem- designated). A Hausdorff space that provides a counter-example to that is, a normal, non-countable para compact Hausdorff space is a Dowker room ( english Dowker space called). The American mathematician Mary Ellen Rudin solved Dowker's problem in 1971 to the extent that she was able to construct a Dowker space within the framework of the Zermelo-Fraenkel set theory with axiom of choice ( ZFC ).

literature

• Karol Borsuk: Sur les prolongements des transformations continues . In: Fundamenta Mathematicae . tape 28 , 1937, pp. 203 .
• CH Dowker : On countably paracompact spaces . In: Canadian Journal of Mathematics . tape 3 , 1951, pp. 219-224 ( MR0043446 ).
• Egbert Harzheim : Introduction to combinatorial topology (=  mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).
• Kiiti Morita : On generalizations of Borsuk's homotopy extension theorem . In: Fundamenta Mathematicae . tape 88 , 1975, pp. 1-6 ( MR0375220 ).
• Gregory Naber: Set-theoretic Topology. With Emphasis on Problems from the Theory of Coverings, Zero Dimensionality and Cardinal Invariants . University Microfilms International, Ann Arbor MI 1977, ISBN 0-8357-0257-X .
• Jun-iti Nagata : Modern General Topology (=  North Holland Mathematical Library . Volume 33 ). 2nd revised edition. North-Holland Publishing, Amsterdam / New York / Oxford 1985, ISBN 0-444-87655-3 ( MR0831659 ).
• Elliott Pearl: Open Problems in Topology II . Elsevier, Amsterdam (et al.) 2007, ISBN 978-0-444-52208-5 , pp. 233-211 .
• Horst Schubert : Topology . 4th edition. BG Teubner Verlag, Stuttgart 1975, ISBN 3-519-12200-6 ( MR0423277 ).
• Michael Starbird: The Borsuk homotopy extension theorem without the binormality condition . In: Fundamenta Mathematicae . tape 87 , 1975, pp. 207-211 ( MR0372810 ).
• Mary Ellen Rudin: A normal space X for which X × I is not normal . In: Fundamenta Mathematicae . tape 73 , 1971, p. 179-186 ( MR0293583 ).
• Stephen Willard: General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley, Reading MA (et al.) 1970 ( MR0264581 ).

References and footnotes

1. ↑ Stephen Willard: General Topology. 1978, p. 155
2. ↑ ^{a b} Gregory Naber: Set-theoretic Topology. 1978, p. 184
3. ↑ Horst Schubert: Topology. 1975, pp. 90, 98
4. ↑ Stephen Willard: General Topology. 1978, p. 147
5. ^ ^{A b} Clifford Hugh Dowker: On countably paracompact spaces. In: Canadian J. Math. 3, pp. 219-224
6. ↑ Stephen Willard: General Topology. 1978, p. 157
7. ↑ Gregory Naber: Set-theoretic Topology. 1978, p. 185
8. ^ ^{A b c} Paul J. Szeptycki: Small Dowker spaces. In: Elliott Pearl (Ed.): Open Problems in Topology II. , Pp. 233–239 ( sciencedirect.com )
9. ↑ Karol Borsuk: Sur les prolongements des transformations continues. In: Fund. Math. 28, p. 203
10. ↑ Egbert Harzheim: Introduction to combinatorial topology. 1978, p. 199
11. ^ Kiiti Morita: On generalizations of Borsuk's homotopy extension theorem. In: Fund. Math. 88, pp. 1-6
12. ↑ Michael Starbird: The Borsuk homotopy extension theorem without the binormality condition. In: Fund. Math. , 87, pp. 207-211
13. ↑ Egbert Harzheim: Introduction to combinatorial topology. 1978, p. 200
14. ↑ ^{a b} Stephen Willard: General Topology. 1978, p. 158
15. ↑ Gregory Naber: Set-theoretic Topology. 1978, pp. 207-227
16. ↑ Jun-iti Nagata: Modern General Topology. 1985, p. 214