List questions from metric-spaces

Published on 08 Nov 2018 - 2:37

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Lindelöf and separable metric space

Published on 25 Mar 2026 - 23:38

75

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Find a continuous bijective function $f: [0,1) \to S^1$ such that $f^{-1}$ is not continuous

Published on 08 Nov 2018 - 6:41

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Is $\Bbb Q$ intersect $[0,1]$ in $\Bbb R$ compact?

Published on 29 Mar 2026 - 20:45

232

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Show that the space of increasing, bounded function is not totally bounded w.r.t. $\sup$-norm

Published on 08 Nov 2018 - 18:03

664

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Show that the support function of a bounded set is continuous.

Published on 26 Mar 2026 - 14:28

147

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Can we define metric derivative on discrete metric space?

Published on 09 Nov 2018 - 5:49

698

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Showing $e(a,b)=\{\min(1,d(a,b))\}$ is equivalent to $d(a,b)$

Published on 25 Mar 2026 - 8:07

400

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Proving $d((x_1,y_1),(x_2,y_2))=\max\{d_1(x_1,x_2),d_2(y_1,y_2)\}$ and $e((x_1,y_1),(x_2,y_2))=d_1(x_1,x_2)+d_2(y_1,y_2)$ are equivalent

Published on 09 Nov 2018 - 19:26

88

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Show that supremum of over a bounded set is continuous.

Published on 09 Nov 2018 - 20:50

484

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Can the constant $3$ in the Vitali covering lemma be replaced by any positive constant less than that in the finite case?

Published on 10 Nov 2018 - 8:53

5.8k

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Proving every metrizable space is normal space

Published on 10 Nov 2018 - 15:31

62

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Delone sets in $\mathbb{R}$ are bi-Lipschitz equivalent

Published on 25 Mar 2026 - 14:39

246

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Space is both $T_1$-space and normal space is Hausdorff space

Published on 10 Nov 2018 - 18:31

51

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Metric space problem (Triangle inequality)

Published on 10 Nov 2018 - 18:43

List Question

Equicontinuous family of functions

Lindelöf and separable metric space

Find a continuous bijective function $f: [0,1) \to S^1$ such that $f^{-1}$ is not continuous

Is $\Bbb Q$ intersect $[0,1]$ in $\Bbb R$ compact?

Show that the space of increasing, bounded function is not totally bounded w.r.t. $\sup$-norm

Show that the support function of a bounded set is continuous.

Can we define metric derivative on discrete metric space?

Showing $e(a,b)=\{\min(1,d(a,b))\}$ is equivalent to $d(a,b)$

Proving $d((x_1,y_1),(x_2,y_2))=\max\{d_1(x_1,x_2),d_2(y_1,y_2)\}$ and $e((x_1,y_1),(x_2,y_2))=d_1(x_1,x_2)+d_2(y_1,y_2)$ are equivalent

Show that supremum of over a bounded set is continuous.

Can the constant $3$ in the Vitali covering lemma be replaced by any positive constant less than that in the finite case?

Proving every metrizable space is normal space

Delone sets in $\mathbb{R}$ are bi-Lipschitz equivalent

Space is both $T_1$-space and normal space is Hausdorff space

Metric space problem (Triangle inequality)

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