Timeline for Cardinalities larger than the continuum in areas besides set theory
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 20, 2021 at 11:01 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
a minor typo
|
| Jun 21, 2016 at 0:50 | comment | added | Joseph Van Name | Therefore, the mapping $f:X\rightarrow\omega$ extends to a continuous surjection $\overline{f}:\beta X\rightarrow\beta\omega$. Therefore, $|\beta X|\geq|\beta\omega|$, so $|\beta X|=2^{2^{\aleph_{0}}}$. | |
| Jun 21, 2016 at 0:47 | comment | added | Joseph Van Name | Emil Jeřábek. If $X$ is a countable completely regular space which is not compact, then $\beta X$ has cardinality $2^{2^{\aleph_{0}}}$. If $X$ is countable and completely regular but not compact, then there is a countable clopen cover $(C_{n})_{n\in\omega}$ which has no finite subcover. Therefore, let $D_{n}=C_{n}\setminus(C_{0}\cup...C_{n-1})$. Then $(D_{n})_{n\in\omega}$ is a partition of $X$ into countably many sets. Without loss of generality, assume that each $D_{n}$ is non-empty. Then define a mapping $f:X\rightarrow\omega$ by $f[D_{n}]=\{n\}$. Then $f$ is a continuous surjection | |
| Aug 8, 2013 at 0:00 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
deleted 1 characters in body
|
| Jun 2, 2011 at 14:05 | comment | added | Emil Jeřábek | The discrete countable space has Stone–Čech compactification of size $2^{2^{\aleph_0}}$. However, there are many countable spaces whose $\beta$ is much smaller. For the extreme case, if $X$ is a countable compact space, then $\beta X=X$ is countable. | |
| Nov 3, 2010 at 18:39 | history | edited | Rachid Atmai | CC BY-SA 2.5 |
added 10 characters in body
|
| Nov 3, 2010 at 18:39 | comment | added | Rachid Atmai | You're right I mean $X$ not $\aleph_0$, let me edit that. | |
| Nov 3, 2010 at 18:32 | comment | added | Todd Trimble | Only the first has size provably greater than the continuum (and you probably mean $2^{2^{|X|}}$. Since Daniel asked for analysis applications, it may be worth pointing out that nonstandard analysis calls for ultrapowers, which in turn calls for a set of nonprincipal ultrafilters such as $\beta(\mathbb{N})$. | |
| Nov 3, 2010 at 18:14 | history | edited | Rachid Atmai | CC BY-SA 2.5 |
added 4 characters in body
|
| Nov 3, 2010 at 17:48 | history | answered | Rachid Atmai | CC BY-SA 2.5 |