Question

First countability of $[0,1]^\mathbb{R}$

score 142 · Answer 1 · 2021-02-16 00:43:07

142

Views

First countability of $[0,1]^\mathbb{R}$

Published on 16 Feb 2021 - 0:43

score 96 · Answer 2 · 2021-02-24 02:41:03

96

Views

Find a non-constant $f_n : \mathbb{R} \rightarrow \mathbb{R}$ so $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$ is continuous.

Published on 24 Feb 2021 - 2:41

score 277 · Answer 3 · 2021-03-17 05:55:47

277

Views

Proving $[0,1]^{\omega}$ in box topology is not compact

Published on 17 Mar 2021 - 5:55

score 35 · Answer 4 · 2021-04-12 16:14:45

35

Views

$X=\prod_{j\in J}X_j$ is Hausdorff(regular) iff each $X_j$ is Hausdorff(regular) on box-topology. Then what about the $T_1$ and product-topology?

Published on 12 Apr 2021 - 16:14

score 366 · Answer 5 · 2021-05-08 22:52:45

366

Views

Box topology and connectedness property

Published on 08 May 2021 - 22:52

score 538 · Answer 6 · 2021-06-24 03:13:20

538

Views

Countable product of $\mathbb{R}$ is not connected with respect to box topology

Published on 24 Jun 2021 - 3:13

score 139 · Answer 7 · 2021-07-22 06:09:36

139

Views

Let $[0, 1]^{\mathbb N}$ be equipped with the box topology.

Published on 22 Jul 2021 - 6:09

score 201 · Answer 8 · 2021-09-27 06:47:40

201

Views

Connectedness in box topology

Published on 27 Sep 2021 - 6:47

score 95 · Answer 9 · 2021-11-22 12:18:09

95

Views

Show that a set of sequences is open and closed on box topology

Published on 22 Nov 2021 - 12:18

score 223 · Answer 10 · 2022-01-08 00:46:59

223

Views

$\prod_{n=1}^{\infty}{\mathbb{R}}$ endowed with the box topology is not first countable.

Published on 08 Jan 2022 - 0:46

score 101 · Answer 11 · 2022-01-11 20:55:25

Let $X=\Bbb R^{\Bbb N}$ be equipped with the box topology. Let $A \subset X$ be the set of bounded sequences. Show that $A$ is clopen.

score 96 · Answer 12 · 2022-04-10 06:08:51

96

Views

Where does this "proof" that $\Bbb{R}^\omega$ is normal in the box topology go awry?

Published on 10 Apr 2022 - 6:08

score 330 · Answer 13 · 2022-04-19 02:01:41

330

Views

Can ($X^I$, product topology) and ($X^I$, box topology) be homeomorphic for some nontrivial $X$ and infinite $I$?

Published on 19 Apr 2022 - 2:01

score 70 · Answer 14 · 2022-09-13 10:00:34

70

Views

Show that the set of all bounded sequences in $X=\mathbb{R}^\Bbb N$ with the box topology is an open and closed set.

Published on 13 Sep 2022 - 10:00

score 68 · Answer 15 · 2022-10-24 07:23:05

68

Views

Difficulty in showing continuity of a function in different topologies

Published on 24 Oct 2022 - 7:23

List Question

First countability of $[0,1]^\mathbb{R}$

Find a non-constant $f_n : \mathbb{R} \rightarrow \mathbb{R}$ so $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$ is continuous.

Proving $[0,1]^{\omega}$ in box topology is not compact

$X=\prod_{j\in J}X_j$ is Hausdorff(regular) iff each $X_j$ is Hausdorff(regular) on box-topology. Then what about the $T_1$ and product-topology?

Box topology and connectedness property

Countable product of $\mathbb{R}$ is not connected with respect to box topology

Let $[0, 1]^{\mathbb N}$ be equipped with the box topology.

Connectedness in box topology

Show that a set of sequences is open and closed on box topology

$\prod_{n=1}^{\infty}{\mathbb{R}}$ endowed with the box topology is not first countable.

Let $X=\Bbb R^{\Bbb N}$ be equipped with the box topology. Let $A \subset X$ be the set of bounded sequences. Show that $A$ is clopen.

Where does this "proof" that $\Bbb{R}^\omega$ is normal in the box topology go awry?

Can ($X^I$, product topology) and ($X^I$, box topology) be homeomorphic for some nontrivial $X$ and infinite $I$?

Show that the set of all bounded sequences in $X=\mathbb{R}^\Bbb N$ with the box topology is an open and closed set.

Difficulty in showing continuity of a function in different topologies

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