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

From Maths
Jump to: navigation, search


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]:


Grade: C
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
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


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


  1. Introduction to Topological Manifolds - John M. Lee