Question

A subset of a metric space is open iff it is a union of open neighborhoods.

score 609 · Answer 1 · 2019-01-16 02:13:15

609

Views

A subset of a metric space is open iff it is a union of open neighborhoods.

Published on 16 Jan 2019 - 2:13

score 1.1k · Answer 2 · 2026-03-30 12:02:38

1.1k

Views

Is $ C ^ 1 [ 0 , 1 ] $ compact? Is the closure of the unit ball of $ C ^ 1 [ 0 , 1 ] $ in $ C [ 0 , 1 ] $ compact?

Published on 30 Mar 2026 - 12:02

score 981 · Answer 3 · 2026-04-01 06:31:47

981

Views

Is $d(x,y)=\frac {\|x-y\|} {\sqrt {1+\|x\|^{2}}\sqrt {1+\|y\|^{2}}}$ a metric on a normed linear space?

Published on 01 Apr 2026 - 6:31

score 451 · Answer 4 · 2019-01-16 20:34:15

451

Views

Showing that $f(x)=x$ for at least one point in $[0,1]$

Published on 16 Jan 2019 - 20:34

score 82 · Answer 5 · 2026-03-25 07:54:21

82

Views

Given a set of points X, locate the ball of maximum radius whose interior contains none of the points in X

Published on 25 Mar 2026 - 7:54

score 102 · Answer 6 · 2019-01-17 14:43:31

102

Views

If $X$ is a real normed linear space and $r>0$, then $B_{r}(x_0+y_0)=B_{r}(y_0)+\{x_0\}$ for fixed $x_0,\,y_0\in X.$

Published on 17 Jan 2019 - 14:43

score 74 · Answer 7 · 2019-01-17 15:39:04

74

Views

If $(N,\|.\|)$ is a non trivial vector space with norm, then the diameter of the unit sphere is $\ge2$

Published on 17 Jan 2019 - 15:39

score 25 · Answer 8 · 2019-01-17 20:37:10

25

Views

Verification of metric axiom 3

Published on 17 Jan 2019 - 20:37

score 2.4k · Answer 9 · 2019-01-18 02:04:42

2.4k

Views

Prove the triangle inequality in R^2

Published on 18 Jan 2019 - 2:04

score 460 · Answer 10 · 2019-01-18 06:01:04

460

Views

Show the closure of a subset of a complete metric space is compact.

Published on 18 Jan 2019 - 6:01

score 447 · Answer 11 · 2026-03-29 17:31:27

447

Views

The Domain of a continuous function is an open set

Published on 29 Mar 2026 - 17:31

score 1k · Answer 12 · 2019-01-18 08:21:16

1k

Views

Axioms of a metric space

Published on 18 Jan 2019 - 8:21

score 34 · Answer 13 · 2019-01-18 10:39:52

Proving that $U=\{x=(x_1,\dots,x_n)\in\Bbb R^n : \sum|x_i|<1\}$ is open in $(\Bbb R^n,d)$: Does the metric $d$ matter?

score 591 · Answer 14 · 2019-01-18 11:30:04

591

Views

If $X$ is a real normed linear space, then $B(0,r)=r B(0,1)$

Published on 18 Jan 2019 - 11:30

score 28 · Answer 15 · 2019-01-18 17:53:23

$f_n:M\mapsto M'$ uniformly converges iff $d'(f_{n_k}(x_k),f(x_k))\rightarrow 0\in\Bbb R\forall$ subsequence $f_{n_k}$ and $\forall$ sequence $x_k$

List Question

A subset of a metric space is open iff it is a union of open neighborhoods.

Is $ C ^ 1 [ 0 , 1 ] $ compact? Is the closure of the unit ball of $ C ^ 1 [ 0 , 1 ] $ in $ C [ 0 , 1 ] $ compact?

Is $d(x,y)=\frac {\|x-y\|} {\sqrt {1+\|x\|^{2}}\sqrt {1+\|y\|^{2}}}$ a metric on a normed linear space?

Showing that $f(x)=x$ for at least one point in $[0,1]$

Given a set of points X, locate the ball of maximum radius whose interior contains none of the points in X

If $X$ is a real normed linear space and $r>0$, then $B_{r}(x_0+y_0)=B_{r}(y_0)+\{x_0\}$ for fixed $x_0,\,y_0\in X.$

If $(N,\|.\|)$ is a non trivial vector space with norm, then the diameter of the unit sphere is $\ge2$

Verification of metric axiom 3

Prove the triangle inequality in R^2

Show the closure of a subset of a complete metric space is compact.

The Domain of a continuous function is an open set

Axioms of a metric space

Proving that $U=\{x=(x_1,\dots,x_n)\in\Bbb R^n : \sum|x_i|<1\}$ is open in $(\Bbb R^n,d)$: Does the metric $d$ matter?

If $X$ is a real normed linear space, then $B(0,r)=r B(0,1)$

$f_n:M\mapsto M'$ uniformly converges iff $d'(f_{n_k}(x_k),f(x_k))\rightarrow 0\in\Bbb R\forall$ subsequence $f_{n_k}$ and $\forall$ sequence $x_k$

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