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Real Analysis: Seemingly Obvious Question about Proving that points of S are an open subset of Metric Space E

score 266 · Answer 1 · 2018-02-13 00:05:08

266

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Real Analysis: Seemingly Obvious Question about Proving that points of S are an open subset of Metric Space E

Published on 13 Feb 2018 - 0:05

score 412 · Answer 2 · 2018-02-13 08:56:11

412

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Show that $d(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$ defines a metric on $(0,\infty)$.

Published on 13 Feb 2018 - 8:56

score 58 · Answer 3 · 2018-02-13 09:49:21

58

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If $d$ is a metric on $M$, show that $\left|d(x,z) - d(y,z)\right|\leq d(x,y)$ for any $x,y,z\in M$.

Published on 13 Feb 2018 - 9:49

score 29 · Answer 4 · 2018-02-14 04:41:32

29

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Two metrics in the space of bounded sequences in $\mathbb{C}$.

Published on 14 Feb 2018 - 4:41

score 160 · Answer 5 · 2026-03-29 01:33:40

160

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Problems on Open Balls

Published on 29 Mar 2026 - 1:33

score 3.9k · Answer 6 · 2018-02-14 17:04:26

3.9k

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Is any subset of $\mathbb{Z}$ open?

Published on 14 Feb 2018 - 17:04

score 1.4k · Answer 7 · 2026-03-29 04:44:03

1.4k

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Show that a metric space $(X,d)$ is totally bounded if and only if the completion of $(X,d)$ is compact.

Published on 29 Mar 2026 - 4:44

score 84 · Answer 8 · 2026-03-22 21:11:24

84

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Lattices in $\ell^\infty(X)$

Published on 22 Mar 2026 - 21:11

score 940 · Answer 9 · 2026-03-24 22:06:27

940

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Sketching open balls

Published on 24 Mar 2026 - 22:06

score 173 · Answer 10 · 2026-03-24 22:08:49

173

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Prove that a sequence is Cauchy in (X,d) if and only if it is Cauchy in (X,d')

Published on 24 Mar 2026 - 22:08

score 98 · Answer 11 · 2026-03-26 18:30:11

98

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Why is $\|y\|_1 = \lambda^{-1}\|x\|_1$ given that $y = \lambda - 1x$?

Published on 26 Mar 2026 - 18:30

score 383 · Answer 12 · 2026-03-25 01:21:16

383

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Fixed Point Property for a special space?

Published on 25 Mar 2026 - 1:21

score 458 · Answer 13 · 2018-02-15 18:33:01

458

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Understanding $l_p$ with the metric $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|}$

Published on 15 Feb 2018 - 18:33

score 31 · Answer 14 · 2026-03-25 01:45:18

31

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Questions about $Lip_k[0,1]$

Published on 25 Mar 2026 - 1:45

score 96 · Answer 15 · 2026-03-29 08:18:22

96

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Showing that $U$ open in $(X, d) \Rightarrow U$ open in $(X, d')$

Published on 29 Mar 2026 - 8:18

List Question

Real Analysis: Seemingly Obvious Question about Proving that points of S are an open subset of Metric Space E

Show that $d(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$ defines a metric on $(0,\infty)$.

If $d$ is a metric on $M$, show that $\left|d(x,z) - d(y,z)\right|\leq d(x,y)$ for any $x,y,z\in M$.

Two metrics in the space of bounded sequences in $\mathbb{C}$.

Problems on Open Balls

Is any subset of $\mathbb{Z}$ open?

Show that a metric space $(X,d)$ is totally bounded if and only if the completion of $(X,d)$ is compact.

Lattices in $\ell^\infty(X)$

Sketching open balls

Prove that a sequence is Cauchy in (X,d) if and only if it is Cauchy in (X,d')

Why is $\|y\|_1 = \lambda^{-1}\|x\|_1$ given that $y = \lambda - 1x$?

Fixed Point Property for a special space?

Understanding $l_p$ with the metric $\rho(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{|x_k-y_k|}{1+|x_k-y_k|}$

Questions about $Lip_k[0,1]$

Showing that $U$ open in $(X, d) \Rightarrow U$ open in $(X, d')$

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