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Quotient topology

  1. Apr 20, 2013 #1
    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.
  2. jcsd
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