Michael straight

The Michael straight line or Michael line , named after Ernest Michael , is a special topological space considered in the mathematical sub-area of topology . Historically, it was the first example of a normal room whose product with a metric space is not normal again.

definition

The Michael straight line is the set from which the term straight line comes from, together with the topology defined as follows : Open sets are the unions , where ${\displaystyle M}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \tau _{M}}$ ${\displaystyle U\cup I}$

• ${\displaystyle U}$an open set in Euclidean topology , that is, a union of intervals with ${\displaystyle (a,b)}$ ${\displaystyle a,b\in \mathbb {R} }$
• ${\displaystyle I}$any subset of the irrational numbers

is. One shows that this defines a topological space . ${\displaystyle M=(\mathbb {R} ,\tau _{M})}$

properties

• Michael Straight is a normal room, even a paracompact room . ${\displaystyle M=(\mathbb {R} ,\tau _{M})}$
• If the space of the irrational numbers is with the relative Euclidean topology, the product is not normal. E. Michael has shown this in his original work cited below. ${\displaystyle X}$ ${\displaystyle M\times X}$
• The Michael straight line cannot be metrized, because otherwise the above product would also be metrizable and therefore normal.
• Michael straight is not a Lindelöf room . If the rational numbers are counted, then the sets and the one-point sets from all irrational points form an open cover that has no countable partial cover, because the union of a countable subset of the cover sets has at most the Lebesgue measure 4. The question of whether it There are also Lindelof spaces whose product with the space of irrational numbers is not normal, touches the axiomatics of set theory . Lindelöf rooms with this property are called Michael rooms , the Michael straight line is not a Michael room because of the missing Lindelöf property. ${\displaystyle (r_{n})_{n}}$ ${\displaystyle (r_{n}-2^{-n},r_{n}+2^{-n})}$ ${\displaystyle \{x\}}$

Individual evidence

1. ↑ Ernest Michael : The product of a normal space and a metric space need not be normal. In: Bulletin of the American Mathematical Society. Vol. 69, No. 3, 1963, pp. 375-376, doi : 10.1090 / S0002-9904-1963-10931-3 .
2. ^ Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (= BI university paperbacks. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 , task 38.
3. ↑ Jun-iti Nagata : Modern General Topology (= North Holland Mathematical Library. Vol. 33). 2., revised edition. North Holland Publishing Co., Amsterdam et al. 1985, ISBN 0-444-87655-3 , p. 197.
4. ^ Johann Cigler, Hans-Christian Reichel: Topology. A basic lecture (= BI university paperbacks. 121). Bibliographisches Institut, Mannheim et al. 1978, ISBN 3-411-00121-6 , p. 136.
5. ^ L. Brian Lawrence: The influence of a small cardinal on the product of a Lindelöf space and the irrationals. In: Proceedings of the American Mathematical Society. Vol. 110, No. 2, 1990, pp. 535-542, doi : 10.2307 / 2048101 .