Question

$(X,d)$ is separable if and only if $X$ does not contain an uncountable closed discrete subset

score 1.7k · Answer 1 · 2018-07-17 02:44:04

1.7k

Views

$(X,d)$ is separable if and only if $X$ does not contain an uncountable closed discrete subset

Published on 17 Jul 2018 - 2:44

score 215 · Answer 2 · 2018-07-17 18:38:02

215

Views

$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly

Published on 17 Jul 2018 - 18:38

score 143 · Answer 3 · 2018-07-18 05:37:04

143

Views

Need help understanding this question about showing continuity in a metric space.

Published on 18 Jul 2018 - 5:37

score 109 · Answer 4 · 2018-07-18 06:55:31

109

Views

The union of $A$ and $B$ is closed in $X$

Published on 18 Jul 2018 - 6:55

score 212 · Answer 5 · 2018-07-18 06:59:35

$A, B$ be two subsets of a metric space such that $\bar A \cap \bar B = \emptyset$, then prove that $\partial(A\cup B)=\partial A \cup \partial B$

score 4.7k · Answer 6 · 2026-03-25 13:56:18

4.7k

Views

Prove or disprove: If $(x_n)$ is a Cauchy sequence in $X$ and $f$ is a continuous function on $X$, then $(f(x_n))$ is a Cauchy sequence.

Published on 25 Mar 2026 - 13:56

score 39 · Answer 7 · 2018-07-18 18:03:41

In a metric space $(X,d)$, $\partial A=\emptyset\quad\forall A\subseteq X$. Is $(X,d)$ a discrete metric space?

score 751 · Answer 8 · 2026-03-28 23:10:32

751

Views

A subset $A\subset \mathbb R^n$ is compact iff every nested sequence of relatively closed, non-empty subsets of $A$ has non-empty intersection.

Published on 28 Mar 2026 - 23:10

score 350 · Answer 9 · 2018-07-19 15:27:28

350

Views

What's $\mathbb{R}$ doing in the definition of a metric space?

Published on 19 Jul 2018 - 15:27

score 1.9k · Answer 10 · 2026-03-25 12:53:51

1.9k

Views

What is the relationship between a metric tensor and a metric?

Published on 25 Mar 2026 - 12:53

score 48 · Answer 11 · 2026-03-27 02:37:23

48

Views

properties of $\min(x_1...x_n)$

Published on 27 Mar 2026 - 2:37

score 514 · Answer 12 · 2026-03-25 06:08:55

514

Views

Showing an isometry.

Published on 25 Mar 2026 - 6:08

score 2.6k · Answer 13 · 2018-07-20 09:14:16

2.6k

Views

Every compact metric space is complete, proof verification

Published on 20 Jul 2018 - 9:14

score 80 · Answer 14 · 2026-03-28 23:16:38

80

Views

Is there a metric space such that for a sequence of nonempty subsets $K_1 \supset K_2 ....$ the intersection of all subsets is empty?

Published on 28 Mar 2026 - 23:16

score 118 · Answer 15 · 2018-07-20 22:59:25

118

Views

Let $X$ be a compact metric space. Prove that $C[X]$ contains a countable subset that separates points.

Published on 20 Jul 2018 - 22:59

List Question

$(X,d)$ is separable if and only if $X$ does not contain an uncountable closed discrete subset

$f$ is continuous, if $f_n$ continuous and $f_n\to f$ uniformly

Need help understanding this question about showing continuity in a metric space.

The union of $A$ and $B$ is closed in $X$

$A, B$ be two subsets of a metric space such that $\bar A \cap \bar B = \emptyset$, then prove that $\partial(A\cup B)=\partial A \cup \partial B$

Prove or disprove: If $(x_n)$ is a Cauchy sequence in $X$ and $f$ is a continuous function on $X$, then $(f(x_n))$ is a Cauchy sequence.

In a metric space $(X,d)$, $\partial A=\emptyset\quad\forall A\subseteq X$. Is $(X,d)$ a discrete metric space?

A subset $A\subset \mathbb R^n$ is compact iff every nested sequence of relatively closed, non-empty subsets of $A$ has non-empty intersection.

What's $\mathbb{R}$ doing in the definition of a metric space?

What is the relationship between a metric tensor and a metric?

properties of $\min(x_1...x_n)$

Showing an isometry.

Every compact metric space is complete, proof verification

Is there a metric space such that for a sequence of nonempty subsets $K_1 \supset K_2 ....$ the intersection of all subsets is empty?

Let $X$ be a compact metric space. Prove that $C[X]$ contains a countable subset that separates points.

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