## Michele_Laino one year ago Recently I saw some inaccurate statements about a question on topology, so I have written this short paper in order to clarify those wrong statements!

1. Michele_Laino

2. anonymous

good job :)

3. anonymous

Do you mind if I save this on my computer?

4. Michele_Laino

thanks! :) @heretohelpalways

5. Michele_Laino

yes sure! :) @SpooderWoman

6. anonymous

Thanks

7. imqwerty

Nice work @Michele_Laino

8. Michele_Laino

here is my reference: $\Large \begin{gathered} {\text{John M}}{\text{. Lee}} \hfill \\ {\text{Introduction to Topological Manifolds}} \hfill \\ {\text{Springer }}\left( {{\text{first edition}},2000} \right) \hfill \\ \end{gathered}$

9. Michele_Laino

thanks! :) @imqwerty

10. anonymous

hi @Michele_Laino you made a wonderful prove there but i fink it is better to forgive and forget. you a making a big issue out of this and you make me feel i caused every thing

11. Michele_Laino

dear @GIL.ojei as I said you yesterday, you don't have to feel yourself the cause of everything

12. amistre64

topology is not an area i am familiar with, but i did just peruse the other posting :) it was enlightening regardless of who was doling out inaccuracies.

13. Michele_Laino

thanks! for your appreciation @amistre64 :)

14. anonymous

@GIL.ojei It could have been anybody :) I think @Michele_Laino did a great job; she probably took time out of her schedule to write this paper. I think it is awesome.

15. anonymous

correction the "it" in my last reply was supposed to be "the paper"

16. Michele_Laino

thanks again! @heretohelpalways

17. Astrophysics

I do not know topology very well, or any for that matter haha, but I totally agree with amistrte, thanks @Michele_Laino :-)

18. Michele_Laino

thanks! @Astrophysics :)

19. anonymous

lol, it's like you read my posts and then butchered them in an attempt to cloak your incorrect answers. it's okay to be wrong sometimes, you know

20. anonymous

there is nothing in this post aside from an unrelated, unnecessary, and pointless proof that singletons are closed in a Hausdorff space that I did not explicitly state in my posts in the original thread. once again, it was made explicitly clear that the definition of an open ball centered at $$p$$ in a metric space $$(X,d)$$ is for $$r>0$$, and so the question was nonsense to start with. furthermore, generalizing it to $$r=0$$ in the immediately obvious way does not yield a singleton set but instead an empty set, and this amounts to the fact that $$d\ge0\implies d\not<0$$, so no points could satisfy the open-ball membership condition $$d(p,x)< 0$$ over $$x\in X$$

21. anonymous

and perhaps the most depressing part of this entire charade is that you're still stating things that are false despite the best efforts of others to help show you the logical error you're making. consider in your paper extending the definition of an open ball in the obvious way by allowing $$r\ge0$$ with the same definition, $$B_r(p)=\{x\in X:d(p,x)<r\}$$so imagine extending this to $$r=0$$, which gives: $$B_0(p)=\{x\in X:d(p,x)<0\}$$well, since $$d\ge0$$ this condition is never satisfied, meaning $$B_0(p)=\emptyset$$but this is not very useful, and clearly we gain nothing from generalizing the definition like this, but that does not alter the unfortunate fact that you are wrong in stating that the generalized definition yields $$B_0(p)=\{p\}$$, since this is stating: $$p\in\{x\in X:d(p,x)<0\}\\\implies d(p,p)<0\\\implies 0<0$$but this is simply not true.

22. anonymous

also, can you give any logical reason to bring Kant into this other than as an attempt to brag about your high school philosophy credentials? you're not even thinking of the same idea of extension, really; Kant is referring to extension as in the property of having extent, not the notion of extension as generalization

23. Michele_Laino