Consider
\(\rm X_1=\left\{\left(x, \sin \frac{1}{x}\right): 0
X2 = [0,1] × {0}
X3 = {(0,1)} then
1
X1 ∪ X2 ∪ X3 is a connected set
2
X1 ∪ X2 ∪ X3 is a path connected set
3
X1 ∪ X2 ∪ X3 is not a path connected but X1 ∪ X2 is path connected.
4
Option 1 and 2 are true but option 3 is false.