Timeline for How continuous can a bijection between line and plane be?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 2, 2018 at 15:53 | comment | added | Joel David Hamkins | I took $(X^c\times\{0\})^c$ to mean the complement in the square, so of course this covers the rest of the square. | |
| Jul 2, 2018 at 15:50 | comment | added | Mitch | @JoelDavidHamkins that the complement covers the square by definition was not at all obvious to me (it's still not). But assuming it does, I can see the bijection. With all these extra comments, this works, but those were a lot of gaps for me. | |
| Jul 2, 2018 at 15:39 | comment | added | Joel David Hamkins | @Mitch The c means complement, so it covers the square, by definition, because the second piece is the complement of the first piece. The point of James's nice answer is that there is a trivial continuous injection on the complement of the Cantor set, and then one uses the rest, the Cantor set itself, to extend it to a bijection. This is possible because the Cantor set, although measure zero and meager, has size continuum. This makes for a bijection of $[0,1]$ with $[0,1]^2$ that is continuous on a measure 1 open set. | |
| Jul 2, 2018 at 13:44 | comment | added | Mitch | Oh OK. But I don't see how $(X^c\times\{0\})\cup (X^c\times\{0\})^c$ covers all of $[0,1]^2$. | |
| Jul 2, 2018 at 13:24 | comment | added | James | $[0,1] = X \cup X^c$, and $[0,1]^2 = (X^c \times \{0\}) \cup (X^c \times \{0\})^c$ are partitions of $[0,1]$ and $[0,1]^2$ respectively (here $(X^c \times \{0\})^c$ is the complement in $[0,1]^2$). $h$ and $g$ are bijections between corresonding peices, so $f$ is also a bijection. | |
| Jul 2, 2018 at 12:58 | comment | added | Mitch | I'm having trouble seeing how this is a bijection between $[0,1]$ and $[0,1]^2$. Where does $X\times X$ come in? | |
| Jul 2, 2018 at 8:18 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
minor typos
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| Jul 1, 2018 at 21:19 | history | edited | James | CC BY-SA 4.0 |
added 186 characters in body
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| Jul 1, 2018 at 21:10 | history | answered | James | CC BY-SA 4.0 |