Question

Show that a sequence in $(X,d)$ is Cauchy iff it is Cauchy in $(X,\sigma)$.

score 149 · Answer 1 · 2018-02-26 20:05:22

149

Views

Show that a sequence in $(X,d)$ is Cauchy iff it is Cauchy in $(X,\sigma)$.

Published on 26 Feb 2018 - 20:05

score 81 · Answer 2 · 2018-02-27 03:14:33

81

Views

$X$ is separable, then the set of all $\mu$-continuity sets forms an algebra

Published on 27 Feb 2018 - 3:14

score 93 · Answer 3 · 2018-02-27 05:55:45

93

Views

Clarification on Metric Induced by Norm is Symmetric

Published on 27 Feb 2018 - 5:55

score 42 · Answer 4 · 2026-03-27 12:03:23

42

Views

Let $Q$ be the set of all $x \in H$ of the form $x = \sum_1^{\infty} c_n u_n$ for $|c_n| \le \frac 1n$. Show that $Q$ is compact.

Published on 27 Mar 2026 - 12:03

score 85 · Answer 5 · 2018-02-27 21:59:30

85

Views

Problem with Rudin's proof that $\operatorname{diam}\mkern2mu\overline{\mkern-2muE}=\operatorname{diam}\text{ }E$

Published on 27 Feb 2018 - 21:59

score 1k · Answer 6 · 2018-02-28 00:05:24

1k

Views

Show that $Y$ is compact iff every collection of open sets in $X$ that covers $Y$ has a finite subcover.

Published on 28 Feb 2018 - 0:05

score 191 · Answer 7 · 2018-02-28 00:15:41

Prove the existence of an uncountable set of points whose pairwise distances are $>\epsilon$ means a metric space is not separable

score 173 · Answer 8 · 2018-02-28 03:05:22

173

Views

How does the 1-Lipschitz and uniform boundedness matter in establishing this as a metric?

Published on 28 Feb 2018 - 3:05

score 2.5k · Answer 9 · 2026-03-27 18:27:42

2.5k

Views

Let $(X, d)$ be a metric space. Prove that for every $r > 0$ and $x\in X$ that $\overline{\rm S_r(x)} \subseteq B_r(x)$

Published on 27 Mar 2026 - 18:27

score 203 · Answer 10 · 2018-02-28 07:39:59

203

Views

every bijection is an homeomorphism

Published on 28 Feb 2018 - 7:39

score 41 · Answer 11 · 2018-02-28 12:44:09

41

Views

Comparing Topologies of Metric Spaces

Published on 28 Feb 2018 - 12:44

score 3.2k · Answer 12 · 2026-03-25 03:03:13

3.2k

Views

Prove that ℚ is disconnected as a subset of ℝ

Published on 25 Mar 2026 - 3:03

score 883 · Answer 13 · 2026-03-22 17:33:06

883

Views

$\Bbb{R}^\omega$ with the Uniform Metric is not 2nd Countable

Published on 22 Mar 2026 - 17:33

score 263 · Answer 14 · 2026-03-25 01:21:35

263

Views

Criterion equivalent to a sequence being Cauchy?

Published on 25 Mar 2026 - 1:21

score 1.5k · Answer 15 · 2026-03-24 21:59:50

1.5k

Views

Isometry between $C[a,b]$ and $C[0,1]$

Published on 24 Mar 2026 - 21:59

List Question

Show that a sequence in $(X,d)$ is Cauchy iff it is Cauchy in $(X,\sigma)$.

$X$ is separable, then the set of all $\mu$-continuity sets forms an algebra

Clarification on Metric Induced by Norm is Symmetric

Let $Q$ be the set of all $x \in H$ of the form $x = \sum_1^{\infty} c_n u_n$ for $|c_n| \le \frac 1n$. Show that $Q$ is compact.

Problem with Rudin's proof that $\operatorname{diam}\mkern2mu\overline{\mkern-2muE}=\operatorname{diam}\text{ }E$

Show that $Y$ is compact iff every collection of open sets in $X$ that covers $Y$ has a finite subcover.

Prove the existence of an uncountable set of points whose pairwise distances are $>\epsilon$ means a metric space is not separable

How does the 1-Lipschitz and uniform boundedness matter in establishing this as a metric?

Let $(X, d)$ be a metric space. Prove that for every $r > 0$ and $x\in X$ that $\overline{\rm S_r(x)} \subseteq B_r(x)$

every bijection is an homeomorphism

Comparing Topologies of Metric Spaces

Prove that ℚ is disconnected as a subset of ℝ

$\Bbb{R}^\omega$ with the Uniform Metric is not 2nd Countable

Criterion equivalent to a sequence being Cauchy?

Isometry between $C[a,b]$ and $C[0,1]$

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