I want to think about this in $\mathbb{R}^1$ and $\mathbb{R}^2$.
In $\mathbb{R}^1$, isn't every interval a connected set? So if the intersection of two intervals is also an interval, that would be a connected set. If the intersection is a single point, that would also be connected. If the intersection is null, that is also a connected set.
Is it safe to say intersections of connected sets are always connected in $\mathbb{R}^1$?
For $\mathbb{R}^2$, intuitively, I think the answer is no. I'm not sure how to prove it though.
PS
There's a similar question posted before the answer was for $\mathbb{R}^2$. Must the intersection of connected sets be connected?
The answer "Consider the intersection of the line segment and the circle in $\varnothing$" I don't understand what they mean by a circle in $\varnothing$.
Best Answer
In $\mathbb R$, you are correct. Every connected set is an interval, and the intersection of two intervals is a connected set.
Of course, to actually prove that, you have to prove that All connected sets in $\mathbb R$ are intervals, which is not too hard, but also not entirely trivial.
In $\mathbb R^2$, take $$A=\{(x,y)|x^2+y^2=1\}$$ and $$B=\{(x, 0)| x\in\mathbb R\}$$
Now: