A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements

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Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, and let [ilmath](Y,\mathcal{ K })[/ilmath] be a discrete topological space with [ilmath]Y:=\{0,1\}[/ilmath] and [ilmath]\mathcal{K}:=\mathcal{P}(Y)[/ilmath][Note 1] then[1]:


Grade: C
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Easy proof to do, exercise on page 87 in Lee's top. manifolds, but it didn't take me very long. Marked as easy

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See also


  1. Note: [ilmath]\mathcal{P}(Y)=\mathcal{P}(\{0,1\})=\{\emptyset,\{0\},\{1\},\{0,1\}\}[/ilmath]
  2. Recall a map is constant if:
    • [ilmath]\forall p,q\in X[f(p)=f(q)][/ilmath]


  1. Introduction to Topological Manifolds - John M. Lee