Metacompact space

Metacompact spaces are examined in the mathematical sub-area of topology . It is a weakening of the concept of paracompact space . This formation of the term goes back to Richard Arens and James Dugundji or RH Bing , the latter author used the term, which is no longer in use today, point by point paracompact .

definition

A topological space is called metacompact if every open cover has a punctual open refinement , that is:

If there is a family of open sets in topological space with , there is another family of open sets such that ${\displaystyle {\mathcal {U}}=(U_{i})_{i\in I}}$ ${\displaystyle X}$ ${\displaystyle \textstyle X=\bigcup _{i\in I}U_{i}}$ ${\displaystyle {\mathcal {V}}=(V_{j})_{j\in J}}$

• ${\displaystyle X=\bigcup _{j\in J}V_{j}}$, that is, is also open coverage of ${\displaystyle {\mathcal {V}}}$ ${\displaystyle X}$
• ${\displaystyle \forall j\in J:\exists i\in I:V_{j}\subset U_{i}}$, that is, is refinement of ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {U}}}$
• ${\displaystyle \forall x\in X:|\{j\in J;x\in V_{j}\}|<\infty }$, that means is finite, each point lies in at most finitely many coverage sets. ${\displaystyle {\mathcal {V}}}$

Examples and characteristics

• Closed sub-spaces of metacompact spaces are again metacompact.
• Products of metacompact spaces are generally not metacompact again: The Sorgenfrey line is certainly metacompact as even a paracompact space, but the Sorgefrey level with the product topology is not metacompact. ${\displaystyle X}$ ${\displaystyle X\times X}$
• All paracompact spaces, in particular all metric spaces , are metacompact, since locally finite overlaps are apparently also punctiform.
• The Dieudonné plank is metacompact but not paracompact.
• Metacompact spaces are orthocompact .
• While paracompact rooms are always normal , this generally does not apply to metacompact rooms, here too the Dieudonné plank can be used as an example.
• Normal, metacompact spaces are countably paracompact.
• There are normal, metacompact rooms that are not paracompact.

Countable metacompact spaces

A topological space is called countable metacompact if every countable , open cover has a punctual open refinement .

This is obviously a weakening of the concept of metacompact space, because the defining property is only required here for countable coverages.

It follows directly from the definitions that countably metacompact Lindelöf spaces are metacompact, conversely, separable , metacompact spaces are Lindelof spaces.

Individual evidence

1. ^ Klaas Pieter Hart , Jun-iti Nagata , Jerry E. Vaughan: Encyclopedia of General Topology. Elsevier-Verlag, 2004, ISBN 0-444-50355-2 , p. 199: Generalizations of Paracompactness
2. ^ HJ Kowalski: Topological spaces. Springer, 1961, Definition 13f, p. 97.
3. ↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 84.
4. ^ J. Dieudonné : Une généralisation des espaces compacts. In: J. Math. Pure Appl. Volume 23, 1944, pp. 65-76.
5. ↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 89.
6. ↑ K. Morita: Star-finite coverings and the start-finite property. In: Math. Japon. Volume 1, 1948, pp. 60-68.
7. ↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , example 143.
8. ↑ Lynn Arthur Steen, J. Arthur Seebach: Counterexamples in Topology. Springer-Verlag, 1978, ISBN 3-540-90312-7 , p. 24.