Suppose an infinite set with cofinite topology were disconnected. Then there are open subsets such that and . In particular, we have that both and are open in the co-finite topology. Is that possible?
Does this question even make sense?Prove that any infinte set with co-finite topology is connected.
Let be an infinite space with co-finite topology.
Let be a two point discrete space where .
Let be any continuous function.
I need to show that f is constant.
My big problem is this:
We know and are open.
Hence .
However, is finite so there exist finitely many points where or where .
By that reasoning, we have a finite number of points where the function is not defined. These points are in so the function is discontinuous at each .
Why is this right or wrong?
The cofinite topology is a neat one in the sense that it cannot support two disjoint open sets. Thus it is also, in fact, a space which is not Hausdorff. Also, it is clear that since every totally disconnected space is Hausdorff that a cofinite space is also not totally disconnected (this was immediately obvious before actually doing the question was my point)
A more interesting question you may want to ask yourself: is a cofinite space every locally connected?