If A is countable subset of $\mathbb{R}^2$ , then $\mathbb{R}^2 - A$ is pathwise connected.
16517
Arcwise connected part of $\mathbb R^2$
136106
Is every open subset of $ \mathbb{R} $ uncountable?
44704
familly of Radially open sets in $\mathbb{R}^2$
89264
Connectedness in the Quotient Space
35069
Connectedness of the boundary
96781
connected manifolds are path connected
135783
Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path connected?
116866
$A\times B$ connected component implies $A,B$ connect components
158189
If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected
47251
Non-compactness of $\mathbb{R}$ with the cocountable topology
147136
Subset of a countable set is itself countable
72556
Is the closure of path-connected set path connected?
34998
How to prove that $\mathbb{Q}$ ( the rationals) is a countable set
31266
Proof of "the continuous image of a connected set is connected"
151719
Connected set in $\mathbb{S}^1$
149317
An open subset of $\mathbb{R}$ is an at-most-countable union of disjoint open intervals
115695
If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected