Proving Proposition: Quotient Space Rn/F is First Countable

In summary, the conversation discusses the proposition that the quotient space Rn/F is first countable if and only if the boundary of F is bounded in Rn. The discussion involves considering the definitions of a first countable space and the boundary of a set, and using the definition of the quotient topology to prove the proposition. The conversation also addresses the need for hints and clarification in solving the problem.
  • #1
hedipaldi
210
0
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?
 
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  • #2
hi hedipaldi! :smile:
hedipaldi said:
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)
 
  • #3
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.
 
  • #4
hedipaldi said:
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​
 
  • #5
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)
 
  • #6
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.
 
  • #7
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.
 
  • #8
Moved material from a separate but related thread into this one.
hedipaldi said:
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
 

What is a quotient space?

A quotient space is a mathematical concept used in topology to describe a space that is obtained from another space by identifying or 'gluing' certain points together. This is done in a way that preserves the topological properties of the original space.

What does it mean for a quotient space to be first countable?

A space is said to be first countable if every point in the space has a countable local base, meaning that there exists a countable collection of open sets that contain the point and any neighborhood of the point contains at least one of these sets.

Why is proving the first countability of a quotient space important?

The first countability of a quotient space is important because it allows us to use sequences to characterize the topology of the space. This is useful in many areas of mathematics, including analysis and differential geometry.

What is the process for proving that a quotient space is first countable?

To prove that a quotient space is first countable, we must show that for every point in the space, there exists a countable collection of open sets that contain the point and any neighborhood of the point contains at least one of these sets. This can often be done by constructing a specific local base for each point in the space.

What are some real-world applications of quotient spaces?

Quotient spaces are used in many areas of mathematics, such as topology, differential geometry, and functional analysis. They are also used in physics to describe the behavior of particles in a space with certain symmetries. In computer science, quotient spaces are used in machine learning algorithms to classify data points into different groups.

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