Timeline for Is a cofinite topology for a set with cardinality between $\aleph_{0}$ and $2^{\aleph_{0}}$ path-connected?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 6, 2019 at 14:01 | vote | accept | Noam Zimhoni | ||
| Jul 6, 2019 at 6:09 | answer | added | Henno Brandsma | timeline score: 5 | |
| Jul 6, 2019 at 5:45 | comment | added | Emil Jeřábek | @WillBrian In fact, this is if and only if, isn't it? | |
| Jul 5, 2019 at 20:40 | comment | added | Wojowu | @PietroMajer I only know that your first statement is true for Hausdorff spaces. The line with origin doubled is T1 and path-connected, but not arc-connected. | |
| Jul 5, 2019 at 20:22 | comment | added | Pietro Majer | A path connected space is also connected by injective paths, so if it has more than one point, it has at least c points (but is it true for T1spaces ?) | |
| Jul 5, 2019 at 20:08 | comment | added | Will Brian | @YCor: The space in question isn't Hausdorff, though. In fact, I think that the cofinite topology on some cardinal $\alpha$ should be path connected provided that $\alpha$ is $\geq$ the cardinal $\acute{\mathfrak{n}}$ from this question: mathoverflow.net/questions/285780/…. The idea is that if $\{X_\xi \,:\, \xi < \alpha \}$ is a partition of $[0,1]$ into closed sets, then the mapping that sends $X_\xi$ to $\xi$ is continuous (when $\alpha$ has the cofinite topology). | |
| Jul 5, 2019 at 19:50 | comment | added | YCor | (No need of compactness in the last sentence of my last comment: the consequence is: every Hausdorff space of cardinal $<c$ is totally path-disconnected.") | |
| Jul 5, 2019 at 19:21 | comment | added | YCor | It's classical that every nonempty Hausdorff compact perfect space has cardinal $\ge c$. The image of a non-constant path in a Hausdorff compact space satisfies these assumptions, and hence has cardinal $\ge c$. Hence, every Hausdorff compact space of cardinal $<c$ is totally path-disconnected. | |
| Jul 5, 2019 at 18:15 | review | First posts | |||
| Jul 5, 2019 at 18:15 | |||||
| Jul 5, 2019 at 18:13 | history | asked | Noam Zimhoni | CC BY-SA 4.0 |