query: If $A$ countable then $\Bbb{R}^2\setminus A$ is path connected
positive_passages.docid: 16517
positive_passages.text: Arcwise connected part of $\mathbb R^2$
negative_passages.docid: 2575
negative_passages.text: Showing that $\mathbb{R}$ is connected
negative_passages.docid: 138202
negative_passages.text: Every open set in $\mathbb{R}$ is a countable union of closed sets
negative_passages.docid: 62793
negative_passages.text: $\mathbb{R}$ with the lower limit topology is not second-countable
negative_passages.docid: 103769
negative_passages.text: The Cartesian product of a finite number of countable sets is countable
negative_passages.docid: 115695
negative_passages.text: If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected
negative_passages.docid: 23225
negative_passages.text: Union of connected subsets is connected if intersection is nonempty
negative_passages.docid: 152442
negative_passages.text: An annulus in $\mathbb R^2$ is path connected
negative_passages.docid: 27253
negative_passages.text: Path connectedness of the set $\{(x,y):(x+1)^{2}+y^{2}\leq 1\}\cup\{(x,y):y=x\sin(\frac{1}{x}),x>0\}$
negative_passages.docid: 30079
negative_passages.text: Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?
negative_passages.docid: 95747
negative_passages.text: Number of countable subsets of $\mathbb{R}$
negative_passages.docid: 114990
negative_passages.text: Is the complement of countably many disjoint closed disks path connected?
negative_passages.docid: 133266
negative_passages.text: If $(X, \mathcal{T})$ has a countable subbasis, then it has a countable basis
negative_passages.docid: 144383
negative_passages.text: The closure of a connected set in a topological space is connected
negative_passages.docid: 18223
negative_passages.text: Prove that the union of two disjoint countable sets is countable
negative_passages.docid: 27937
negative_passages.text: Cartesian Product of Two Countable Sets is Countable
negative_passages.docid: 3649
negative_passages.text: Closed unit interval is connected proof