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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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- Thread starter hedipaldi
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- #1

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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?

- #2

tiny-tim

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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

which one can you do?

(start by setting out the definitions)

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- #4

tiny-tim

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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

try now

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- #6

Mark44

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Mark44

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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

Mark44

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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

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