What happens to the connectedness of $\mathbb R^2$ when countable many points are removed? Does $\mathbb R^2$ remain connected when countably many points are removed? Does it remain path connected?
This is not homework but is in response to working several problems where countable or uncountable many points are removed and the pattern always holds.
Thank you.
 A: Hint: how many lines go through a given point?
Namely, the cardinality of the set of lines going through a given point $(a,b)$ is uncountable.  To see this, we can assume for simplicity that $(a,b)=(0,0)$, so every line goes through the origin.  Then we can look at the unit circle centered at the origin, where a line has the form $\{(x,y) \mid y=ax\}$ for some $a$.  The claim is that this set is uniquely defined by $a$, in the sense that if $a\neq b$, then $\{(x,y) \mid y=ax\}\neq \{(x,y) \mid y=bx\}$.  To see this, note that if $x\neq 0$, then given $(x,y)$ in both sets, we find that $ax=y=bx$.  Since $x\neq 0$, we can divide $x$ on both sides to find that $a=b$, contrary to assumption.  Thus, $(0,0)$ is the only point in common, and the two sets cannot be equal.  Thus, we see that the set of lines is in bijection with $\mathbb{R}$.
From here, the idea is that given two distinct points $(a,b)$ and $(c,d)$ that are not in some countable subset $A$ of $\mathbb{R}$, we first look at all the lines passing through $(a,b)$ and all the lines passing through $(c,d)$.  To each point in $A$ is there is exactly one line in $\mathbb{R}^2$ passing through both it and $(a,b)$, and similarly with $(c,d)$.  Since $A$ is countable, the image of this assignment is countable, and it follows that there are uncountably many lines passing through each of $(a,b)$ and $(c,d)$ but not containing any point of $A$; call the sets of such lines $L_1$ and $L_2$, respectively.  Thus, we can certainly pick a line $\ell_1\in L_1$ going through $(a,b)$.  If this line also contains $(c,d)$, then we're done.  Otherwise, there is only one line $\ell$ going through $(c,d)$ that does not intersect $(a,b)$ (i.e. the one parallel to $\ell_1$ that goes through $(c,d)$), so we just pick any line $\ell_2\in L_2\setminus \{\ell\}$.  
Use these lines to develop the continuous curve that goes from $(a,b)$ to $(c,d)$.
