Proving Proposition: Quotient Space Rn/F is First Countable

[hedipaldi]

Hi,
I am trying to prove the following proposition:
Let F be a closed subset of the Euclidean space Rn.Then the quotient space Rn/F is first countable if and only if the boundary of F is bounded in Rn.
Any ideas?

 

[tiny-tim]

hi hedipaldi!

Let F be a closed subset of the Euclidean space Rn.Then the quotient space Rn/F is first countable if and only if the boundary of F is bounded in Rn.
Any ideas?

"if and only if", so that's two proofs …

which one can you do?​

(start by setting out the definitions)

 

[hedipaldi]

you say that one side follows directly from the definitions?I am not so experienced in dealing with quotient spaces (though i am familiar with the definitions) so i really need some hints.

 

[tiny-tim]

you say that one side follows directly from the definitions?

no, I'm saying that you haven't shown any work, you haven't used the homework template, and writing out the definitions would be a good way to start

try now​

 

[hedipaldi]

The definitions are not the problem,i have them in front of me. I just need some hints (I am not a student and this is not homework,i am only curious to solve it)

 

[Mark44]

tiny-tim has given you a hint - start writing some definitions, such as those for a first countable space and the boundary of a set.

 

[Mark44]

I am moving this thread to Homework & Coursework section.

From the PF Rules:

Any and all high school and undergraduate homework assignments or textbook style exercises for which you are seeking assistance are to be posted in the appropriate forum in our Homework & Coursework Questions area--not in blogs, visitor messages, PMs, or the main technical forums. This should be done whether the problem is part of one's assigned coursework or just independent study.

 

[Mark44]

Moved material from a separate but related thread into this one.

i used the definition of the quotient topology,considered a countable
basis at F in the quotient space and it's inverse open sets in Rn
under the natural map.I fail to see the point here and need someolution