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    Connected normal and regular spaces with more than one point are uncountable

    E-YT6Q8

    Area
    Topology
    Topics
    Connectedness, Separation Axioms
    Status
    draft

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

    2. Show that a connected regular space having more than one point is uncountable. [Hint: Any countable space is Lindelöf.]

    Solution

    1. Let \(X\) be connected and normal, with distinct points \(a,b\). Since normal spaces are \(T_1\), the singleton sets \(\{a\},\{b\}\) are closed. By Urysohn’s lemma there is a continuous function \[ f:X\to[0,1] \] with \(f(a)=0\) and \(f(b)=1\). The image \(f(X)\) is connected and contains \(0\) and \(1\), so \[ f(X)=[0,1]. \] Thus \(X\) surjects onto an uncountable set and is therefore uncountable.

    2. Let \(X\) be connected, regular, and have more than one point. If \(X\) were countable, then it would be Lindelöf. A regular Lindelöf space is normal by §32. Part (a) would then imply \(X\) is uncountable, contradiction. Hence \(X\) is uncountable.

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