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    Connected metric spaces with more than one point are uncountable

    E-RZLC7

    Area
    Topology
    Topics
    Connectedness, Metric Spaces
    Seen at
    WESLEYAN
    Years
    2011
    Status
    draft

    Show that a connected metric space having more than one point is uncountable.

    Solution

    Let \(X\) be connected and let \(a\ne b\) in \(X\). Put \(r=d(a,b)>0\). The function \[ f:X\to\mathbb R,\qquad f(x)=d(a,x) \] is continuous. Since \(X\) is connected, \(f(X)\) is connected in \(\mathbb R\), hence an interval. It contains \(0=f(a)\) and \(r=f(b)\), so \[ [0,r]\subset f(X). \] Thus there is a surjection from \(X\) onto the uncountable set \([0,r]\). A countable set cannot surject onto an uncountable set. Therefore \(X\) is uncountable.

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