Question

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!

Answer 1

good job :)

Answer 2

Do you mind if I save this on my computer?

Answer 3

thanks! :) @heretohelpalways

Answer 4

yes sure! :) @SpooderWoman

Answer 5

Thanks

Answer 6

Nice work @Michele_Laino

Answer 7

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

Answer 8

thanks! :) @imqwerty

Answer 9

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

Answer 10

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

Answer 11

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.

Answer 12

thanks! for your appreciation @amistre64 :)

Answer 13

@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.

Answer 14

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

Answer 15

thanks again! @heretohelpalways

Answer 16

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

Answer 17

thanks! @Astrophysics :)

Answer 18

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

Answer 19

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 \(x\in X\)

Answer 20

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.

Answer 21

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

Answer 22

From the last reply to my post, I see that other details are needed:
saying that an open ball can have radius r=0, is how to make an illogical statement, since, as I wrote before, an open ball with radius r=0 doesn't exist, at least in my books, maybe that into your textbooks an openball with radius r=0 does exist, I don't know.

I never said that 0<0, I always said, and I say it here again, that the claim that an open ball can have radius r=0, is simply an absurd statement without any logical foundation, so why to attempt to solve a question which has not solution?

Again, someone want to accuse me of bragging myself, of course that is false, here is why:
mathematics is the natural subject of application of the Kantian metaphysics, infact all mathematics is built using synthetic judgements a priori. In that sense a generalization is an enlargement of the logical validity of a judgement, at the opposite end, saying that the definition of an openball can be made also with r=0, means that I'm doing an enlargement of an empiricism, and, as Kant has said, an enlargement of the empiricism can not never become a synthetic judgement a priori.
As I can see, the Kantian thought has not been well understood!
Furthermore, I have not attended a high school, since I have attended a vocational school, in other words, I learned the kantian philosophy at university.

Finally, another note for my very arrogant interlocutor:
my exposition, is not a charade, it is a collection of some proofs which your friend requested from me (please read the replies inside the previous post).

Again I see that your arrogance and rudeness are higher than your knowledge.