Timeline for Metric spaces containing a topological disc
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2023 at 21:47 | comment | added | Moishe Kohan | @IgorBelegradek: I think "A useful functor and three famous examples in topology" by Williams or "Degree-one, monotone self-maps of the Pontryagin surface are near-homeomorphisms" by Daverman and Thickstun. | |
| Apr 24, 2023 at 20:36 | comment | added | Paul Fabel | If $X$ is metric then $X$ contains a disk iff there exists a map from $R^{2}$ onto a subspace $Y$ of $X$, such that compacta in $Y$ have compact preimage, and so that the preimage of each $y$ in $Y$ does not separate the plane. | |
| Apr 24, 2023 at 20:20 | comment | added | Paul Fabel | Iff there exists a quotient map from $R^{2}$ onto a subspace of $X$ such that each point preimage is a non separating continuum, and so that the point preimages form an upper-semi-continuous decomposition of the plane. There is some redundancy in this answer if $X$ has a reasonable topology. | |
| Apr 24, 2023 at 18:43 | comment | added | Igor Belegradek | @MoisheKohan: I gather a Pontryagin surface is a certain connected sum of infinitely many projective planes, and relevant properties are that it is 2-dimensional, homogeneous and not locally contractible (hence cannot contain a 2-disk). What is a good reference for the properties? | |
| Apr 24, 2023 at 18:02 | comment | added | Moishe Kohan | Yes, this is insufficient, Pontryagin surface would be an example. | |
| Apr 24, 2023 at 16:15 | history | asked | Jeremy Brazas | CC BY-SA 4.0 |