Timeline for Topological spaces that resemble the space of irrationals
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2021 at 16:19 | comment | added | Gabriel Medina | @Noah Schweber Thanks a lot. | |
| Mar 28, 2021 at 19:58 | comment | added | Noah Schweber | @GabrielMedina Let $A=(\alpha_i)_{i\in\mathbb{N}}$ be any sequence of permutations of $\omega$. Then consider the map $$f_A:\omega^\omega\rightarrow\omega^\omega: (c_i)_{i\in\mathbb{N}}\mapsto(\alpha_i(c_i))_{i\in\mathbb{N}}.$$ It's easy to check that this is an autohomeomorphism of $\omega^\omega$. Now given any two elements of Baire space $v=(v_i)_{i\in\mathbb{N}},w=(w_i)_{i\in\mathbb{N}}$, let $\alpha_i$ be the permutation of the naturals which switches $v_i$ and $w_i$ and fixes every other element of $\mathbb{N}$. Then the associated map $F_{(\alpha_i)_{i\in\mathbb{N}}}$ swaps $v$ and $w$. | |
| Feb 27, 2021 at 1:18 | comment | added | Gabriel Medina | Hi Stefan, do you have any idea to prove that $\omega^{\omega}$ is a homogeneous space? | |
| Jul 30, 2010 at 22:57 | history | answered | Stefan Geschke | CC BY-SA 2.5 |