Let ℝ denote the set of all real numbers. Consider the following topological spaces.
X1 = (ℝ, T1), where T1 is the upper limit topology having all sets (a, b] as basis.
X2 = (ℝ, T2), where T2 = {U ⊂ ℝ : ℝ\U is finite} ∪ {Ø}.
Then
1
both X1 and X2 are connected
2
X1 is connected and X2 is NOT connected
3
X1 is NOT connected and X2 is connected
4
neither X1 nor X2 is connected