Every continuous map from a non-empty connected space to a discrete space is constant

Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a non-empty connected topological space and let [ilmath](Y,\mathcal{P}(Y))[/ilmath] be any discrete topological space, then:

• every continuous map, [ilmath]f:X\rightarrow Y[/ilmath] is constant

Proof

Grade: C

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Easy proof to do, I've done it on paper and there is nothing nasty about it

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See also

• 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