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Math Help - Countability in topological spaces

  1. #1
    MHF Contributor Siron's Avatar
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    Countability in topological spaces

    Hi,

    I need to prove the following:
    Consider the topological space \mathbb{R} with the euclidian topology. Define an equivalence relation on \mathbb{R} as follows xRy \Leftrightarrow x=y \ \mbox{or} \ \{x,y\} \subset \mathbb{N}. Now, consider the quotient topology on \mathbb{R}/ R then this space is neither \mbox{C}1(first countable axiom) nor \mbox{C}2 (second countable axiom), because there doens't exist a countable base for the neighbourhoud filter of \overline{0}.

    Can someone explain this for me?

    Thanks!
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    Re: Countability in topological spaces

    How to prove,in topological space,every compact space is separable.?
    Last edited by Plato; December 7th 2012 at 03:43 AM.
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