Timeline for Unnecessary uses of the axiom of choice
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 2, 2022 at 12:31 | comment | added | Z. M | The proof in the book is neat, but it is also not hard to adapt textbook proofs to this case. Indeed, suppose that $f$ is sequentially continuous at $x$. Suppose that there exists $r>0$ such that, for every $n\in\mathbb N$, there exists a point $u\in B(x,1/n)$ such that $d(f(u),f(x))>r$. Then by sequential continuity of $f$ at $u$, there exists a rational number $v\in B(x,1/n)$ such that $d(f(v),f(x))>r/2$, and we can choose $v_n$ to be the greatest rational number $v\in B(x,1/n)$ such that $d(f(v),f(x))>r/2$ with respect to a well-ordering of $\mathbb Q$. | |
| Mar 2, 2022 at 10:10 | comment | added | Martin Sleziak | I have considered mentioning (at least in a comment) that in Herrlich's book Section 3.2 is called Unnecessary Choice. However, the poster of this question excluded topology and the results in this section seem to be related to topology. In any case, the fact that for global continuity we do not need choice is certainly an interesting result - I'll add a link to a related post on Mathematics: Continuity and the Axiom of Choice. | |
| Mar 2, 2022 at 10:06 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added Google Books link
|
| S Mar 2, 2022 at 9:58 | history | answered | Gro-Tsen | CC BY-SA 4.0 | |
| S Mar 2, 2022 at 9:58 | history | made wiki | Post Made Community Wiki by Gro-Tsen |