Question

Is the set of all monic polynomials in $\mathbb{C}[X]$ with roots in a compact set $K$ compact?

score 183 · Answer 1 · 2020-07-03 18:53:06

183

Views

Is the set of all monic polynomials in $\mathbb{C}[X]$ with roots in a compact set $K$ compact?

Published on 03 Jul 2020 - 18:53

score 496 · Answer 2 · 2020-07-04 03:36:37

496

Views

Why does compactness of a subset in a Euclidean space imply that it is closed and bounded?

Published on 04 Jul 2020 - 3:36

score 722 · Answer 3 · 2020-07-04 19:31:04

722

Views

Show that there is a constant $M$ such that for all $x,y \in X$ we have $|f(x) - f(y)| \leq M |x-y| + \epsilon$.

Published on 04 Jul 2020 - 19:31

score 261 · Answer 4 · 2020-07-05 04:07:23

261

Views

Proof that if $(X,d)$ is a metric space and $\forall f:X\to \Bbb R, (f\text{ continuous} \implies f \text{ is bounded})$ then $X$ is compact

Published on 05 Jul 2020 - 4:07

score 49 · Answer 5 · 2020-07-05 16:51:38

49

Views

Prove that $U \backslash f(U)$ is an open set in $X$, where $U=\{x \in X: d(x,x_0)<1\}$.

Published on 05 Jul 2020 - 16:51

score 525 · Answer 6 · 2026-04-03 17:55:24

525

Views

If continuous images of $X$ are closed in every $Y$, is $X$ a compact space?

Published on 03 Apr 2026 - 17:55

score 136 · Answer 7 · 2026-03-26 11:03:42

136

Views

Alexandroff and Stone Čech compactification of topological groups.

Published on 26 Mar 2026 - 11:03

score 83 · Answer 8 · 2020-07-06 22:50:25

83

Views

Definition of compactly supported functions

Published on 06 Jul 2020 - 22:50

score 153 · Answer 9 · 2020-07-07 00:39:08

153

Views

How do I prove that $(X,d)$ is compact.

Published on 07 Jul 2020 - 0:39

score 751 · Answer 10 · 2020-07-07 22:04:19

751

Views

Prove that every set and subset with the cofinite topology is compact

Published on 07 Jul 2020 - 22:04

score 258 · Answer 11 · 2020-07-08 00:15:35

258

Views

Show that there exists a metric $d$ on $\mathbb{R}$ such that $(\mathbb{R},d)$ is compact

Published on 08 Jul 2020 - 0:15

score 52 · Answer 12 · 2026-04-01 23:14:20

52

Views

Compactness and connectedness of the topology having as base $\mathscr{B}=\{ A_{a,b}\mid a,b \in \mathbb{R}\}$

Published on 01 Apr 2026 - 23:14

score 97 · Answer 13 · 2026-03-27 22:18:53

97

Views

Compactness, connecteness and Hausdorffness on $ (S^2 /\mathscr{R}, \tau_{e_{|S^2}}/\mathscr{R}) $

Published on 27 Mar 2026 - 22:18

score 43 · Answer 14 · 2026-03-30 00:22:58

43

Views

Extrema of $\exp(x+y+z)$ on $2$D-submanifold of $\mathbb{R}^3$

Published on 30 Mar 2026 - 0:22

score 329 · Answer 15 · 2020-07-09 17:19:16

329

Views

quotients of locally compact space

Published on 09 Jul 2020 - 17:19

List Question

Is the set of all monic polynomials in $\mathbb{C}[X]$ with roots in a compact set $K$ compact?

Why does compactness of a subset in a Euclidean space imply that it is closed and bounded?

Show that there is a constant $M$ such that for all $x,y \in X$ we have $|f(x) - f(y)| \leq M |x-y| + \epsilon$.

Proof that if $(X,d)$ is a metric space and $\forall f:X\to \Bbb R, (f\text{ continuous} \implies f \text{ is bounded})$ then $X$ is compact

Prove that $U \backslash f(U)$ is an open set in $X$, where $U=\{x \in X: d(x,x_0)<1\}$.

If continuous images of $X$ are closed in every $Y$, is $X$ a compact space?

Alexandroff and Stone Čech compactification of topological groups.

Definition of compactly supported functions

How do I prove that $(X,d)$ is compact.

Prove that every set and subset with the cofinite topology is compact

Show that there exists a metric $d$ on $\mathbb{R}$ such that $(\mathbb{R},d)$ is compact

Compactness and connectedness of the topology having as base $\mathscr{B}=\{ A_{a,b}\mid a,b \in \mathbb{R}\}$

Compactness, connecteness and Hausdorffness on $ (S^2 /\mathscr{R}, \tau_{e_{|S^2}}/\mathscr{R}) $

Extrema of $\exp(x+y+z)$ on $2$D-submanifold of $\mathbb{R}^3$

quotients of locally compact space

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