Can I have help:
let be Uncountable set with cofinite Topology. Show that every continuous function is constant.
Here's the simplest possible way I can think to explain it. Suppose that for some we can then separated the two by open sets (since is Hausdorff) say and . We have then that and are open and non-empty. That said, note that if , but we know that both are finite since the only open sets in are and the complement of finite sets, and so if is infinite then which implies that contradictory to assumption. Thus, is the union of two finite sets, and so finite, but this is ludicrous since their union is .
Note more generally this proves that any map where is infinite and Hausdorff is constant.
Last edited by Drexel28; September 28th 2011 at 04:35 PM.