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I am trying to prove the following property:

Let F be a closed subset of Rn (the n-dimensional euclidean space),and consider the quotient space Rn/F.Then the quotient space satisfies the first countability axiom (i.e there is a countable base at each point) if and only if the boundary of F is bounded in Rn.

The boundary of F is compact in Rn,so taking coverings with shrinkind radii and finite subcoverings might lead to a countable basis of F (F as an element of the quotient space).But i don't know how to proceed.

Thank's for any help.

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# Quotient topology

Can you offer guidance or do you also need help?

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