query: In the connected component of $\{z \in \mathbb{C} : |p(z)| \le 1\}$, $p$ must vanish at least once.
positive_passages.docid: 153107
positive_passages.text: Does every connected component of $\{z : |P(z)|<1 \}$ contain a zero?
negative_passages.docid: 142901
negative_passages.text: "The closure of the unit ball of $C^1[0, 1]$ in $C[0, 1]$" and its compactness
negative_passages.docid: 101286
negative_passages.text: How show that $|a_{n}-1|\le c\lambda ^n,\lambda\in (0,1)$
negative_passages.docid: 67625
negative_passages.text: $C([0, 1])$ is not complete with respect to the norm $\lVert f\rVert _1 = \int_0^1 \lvert f (x) \rvert \,dx$
negative_passages.docid: 102084
negative_passages.text: How to show path-connectedness of $GL(n,\mathbb{C})$
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: 108740
negative_passages.text: $C^1[a,b]$ is closed in $C[a,b]$
negative_passages.docid: 147197
negative_passages.text: Number of connected components of the complement of a compact
negative_passages.docid: 105141
negative_passages.text: $\mathbb R[x]/\langle x^2+1\rangle\simeq\mathbb C$ and $\mathbb R[x]/\langle x^2+1\rangle$ contains a zero of $x^2+1$
negative_passages.docid: 56636
negative_passages.text: A problem of Schwarz-Pick theorem in complex analysis
negative_passages.docid: 12104
negative_passages.text: How to prove this result involving the quotient maps and connectedness?
negative_passages.docid: 136818
negative_passages.text: There is a $C > 0$ such that $\vert f(z)-f(w)\vert \leq C\vert z-w\vert$ for a holomorphic map into $B(0, R)$.
negative_passages.docid: 38185
negative_passages.text: Prove that $\{\cos(nx) : n=1, ... ,\infty \}$ is bounded but not totally bounded in $C([0,\pi])$
negative_passages.docid: 144693
negative_passages.text: Show that if $B \subseteq C$, then $\mathcal{P}(B) \subseteq \mathcal{P}(C)$
negative_passages.docid: 84386
negative_passages.text: Domain strictly contained in the intersection of localizations at the primes of height one
negative_passages.docid: 19754
negative_passages.text: $\mathbb{R}$ \ $\mathbb{Q}$ and $\mathbb{R}^2\setminus\mathbb{Q}^2$ disconnected?
negative_passages.docid: 57669
negative_passages.text: A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.