query: Show that the complement of every connected subset of $S^1$ is also connected.
positive_passages.docid: 151719
positive_passages.text: Connected set in $\mathbb{S}^1$
negative_passages.docid: 104411
negative_passages.text: sum of two connected subset of $\mathbb{R}$
negative_passages.docid: 158850
negative_passages.text: Is it true that a boundary of a simply connected and bounded set is connected in $\mathbb{C}$?
negative_passages.docid: 47056
negative_passages.text: Show that the boundary of a set equals the boundary of its complement
negative_passages.docid: 31266
negative_passages.text: Proof of "the continuous image of a connected set is connected"
negative_passages.docid: 88301
negative_passages.text: What are all of the connected subsets of $\mathbb{Q}$?
negative_passages.docid: 123613
negative_passages.text: Double Complement of a set proof
negative_passages.docid: 109682
negative_passages.text: Proposed proof of: If $A \subset B \subset \bar{A}$ and $A$ is connected, then $B$ is connected
negative_passages.docid: 14048
negative_passages.text: Proof that the complement of zero-measure set on $[0,1]$ is dense in $[0,1]$
negative_passages.docid: 153492
negative_passages.text: Let $p\colon X \to Y$ be a quotient map then if each set $p^{-1}({y})$ is connected, and if $Y$ is connected, then $X$ is connected.
negative_passages.docid: 10541
negative_passages.text: How to prove Mandelbrot set is simply connected?
negative_passages.docid: 4151
negative_passages.text: Every subspace of a vector space has a complement
negative_passages.docid: 45405
negative_passages.text: An elementary way to show any bounded subset of $\Bbb{R}^k$ is totally bounded
negative_passages.docid: 88278
negative_passages.text: Show $S = f^{-1}(f(S))$ for all subsets $S$ iff $f$ is injective
negative_passages.docid: 12104
negative_passages.text: How to prove this result involving the quotient maps and connectedness?
negative_passages.docid: 47680
negative_passages.text: Prove that the union of the interior of a set and the boundary of the set is the closure of the set
negative_passages.docid: 6227
negative_passages.text: Showing diffeomorphism between $S^1 \subset \mathbb{R}^2$ and $\mathbb{RP}^1$