query: Prove that a set A⊂ℝ is connected iff it is an interval.
positive_passages.docid: 156520
positive_passages.text: Intervals are connected and the only connected sets in $\mathbb{R}$
negative_passages.docid: 43535
negative_passages.text: show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
negative_passages.docid: 93771
negative_passages.text: Prove that for any set $A$ there is some $x \notin A$
negative_passages.docid: 124234
negative_passages.text: Prove that if $X$ and it's closure $\overline X$ are connected and if $X\subset Y \subset \overline X$, show that Y is also connected.
negative_passages.docid: 77621
negative_passages.text: Continuity of function mapping connected set to connected set
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: 83665
negative_passages.text: Prove a assertion concerning Lebesgue space
negative_passages.docid: 4602
negative_passages.text: Outline of a proof that $\mathbb{R}^2 - A$ where A is countable is path-connected
negative_passages.docid: 94781
negative_passages.text: Prove that $M(t)=\sup_ {a \leq x \leq t} f(x)$ given $f(x)$ is continuous on $[a,b]$
negative_passages.docid: 23225
negative_passages.text: Union of connected subsets is connected if intersection is nonempty
negative_passages.docid: 136980
negative_passages.text: Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
negative_passages.docid: 136158
negative_passages.text: $X$ has Indiscrete topology $\implies$ any subset of $X$ is connected
negative_passages.docid: 109570
negative_passages.text: How to show that the set of points of continuity is a $G_{\delta}$
negative_passages.docid: 125233
negative_passages.text: Why is an open interval not a compact set?
negative_passages.docid: 154559
negative_passages.text: For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
negative_passages.docid: 112731
negative_passages.text: If $Y$ is connected, why is $A\cup Y$ connected in this case?
negative_passages.docid: 69499
negative_passages.text: Let $A\subset\mathbb{R}$ a measurable and bounded set. Show that exists for each $0\alpha$.