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

From Maths

## Contents

## Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] be a non-empty^{[Note 1]} *connected* topological space and let [ilmath](Y,\mathcal{P}(Y))[/ilmath] be any discrete topological space, then^{[1]}:

- every continuous map, [ilmath]f:X\rightarrow Y[/ilmath] is constant
^{[Note 2]}

## Proof

Grade: C

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The message provided is:

The message provided is:

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

**This proof has been marked as an page requiring an easy proof**## See also

## Notes

- ↑ meaning [ilmath]X\neq\emptyset[/ilmath]
- ↑ It should go without saying, but a map is constant if [ilmath]\forall p,q\in X[f(p)=f(q)][/ilmath]

## References