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

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==Statement==
Let {{Top.|X|J}} be a non-empty<ref group="Note">meaning {{M|1=X\neq\emptyset}}</ref> ''{{link|connected|topology}}'' [[topological space]] and let {{M|(Y,\mathcal{P}(Y))}} be any [[discrete topology|discrete topological space]], then{{rITTMJML}}:
* every [[continuous map]], {{M|f:X\rightarrow Y}} is [[constant map|constant]]<ref group="Note">It should go without saying, but a map is constant if {{M|1=\forall p,q\in X[f(p)=f(q)]}}</ref>
==Proof==
{{Requires proof|grade=C|msg=Easy proof to do, I've done it on paper and there is nothing nasty about it|easy=true}}
==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]]

==Notes==
<references group="Note"/>
==References==
<references/>
{{Theorem Of|Topology}}