I am asked to find a subset $A$ of $\Bbb R^2$ with the following property:
$(1).$ $A$ is connected.
$(2).$ $A-F$ is connected if $F$ is finite.
$(3).$ $A-C$ is disconnected for some countably infinite subset $C\subset A$.
I am beginning this problem with an attempt of setting $A=\{(x,y):x\in\Bbb Q\text{ or }y\in\Bbb Q\}$.
I am guessing $A-\Bbb Q^2$ is disconnected.
In fact, intuitively $A-\Bbb Q^2$ is totally path-disconnected: there could be no path connecting any of the two distinct points in $A-\Bbb Q^2$. Can we deduce $A-\Bbb Q^2$ is disconnected from here?
Fun fact: such space is not homeomorphic to $\Bbb R^2$ as removing countably many points in $\Bbb R^2$ doesn’t affect connectedness.
Best Answer
I feel you are overthinking it. Let me suggest a simpler example: draw a line $y=0$ on your plane, then throw away all non-integer points on the line. What remains is two open half-planes held together with countably many points. This thing is obviously connected and even path-connected, and will remain so if you remove finitely many points.
So it goes.