Timeline for What is this equivalence relation on topological spaces: there are bijective continuous maps in both directions
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 18 at 13:30 | vote | accept | M. Winter | ||
| Feb 17 at 18:51 | comment | added | Will Brian | @HenrikRüping: How about $X = $ the Baire space and $Y = $ a disjoint union of the Baire space and the Cantor space. The Baire space is nowhere compact, so $X$ and $Y$ are not homeomorphic. But there is a continuous bijection $X \rightarrow Y$ (Baire space is homeomorphic to two copies of itself, and you can find a continuous bijection from one of these copies onto Cantor space) and from $Y \rightarrow X$ (roughly, if you excise a copy of Cantor space from Baire space, what's left is Baire space; doing this in reverse gives you the required map). | |
| Feb 17 at 1:02 | answer | added | Ramiro de la Vega | timeline score: 14 | |
| Feb 16 at 23:07 | comment | added | Willie Wong | @SamHopkins: finite CW complexes are compact hausdorff. It is a theorem that if $X$ is compact and $Y$ Hausdorff, then any continuous bijection $\phi:X\to Y$ is a homeomorphism. | |
| Feb 16 at 19:56 | history | became hot network question | |||
| Feb 16 at 16:22 | comment | added | Mark Grant | More examples on MSE: math.stackexchange.com/questions/4461553/… and math.stackexchange.com/questions/4486360/… | |
| Feb 16 at 14:31 | comment | added | Sam Hopkins | Is there an example with finite CW complexes? | |
| Feb 16 at 12:58 | comment | added | HenrikRüping | What is the easiest example of two such nonhomeomorphic spaces ? | |
| Feb 16 at 11:56 | history | asked | M. Winter | CC BY-SA 4.0 |