Question

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

score 143 · Answer 1 · 2026-02-23 04:52:10

143

Views

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

Published on 23 Feb 2026 - 4:52

score 97 · Answer 2 · 2026-02-23 04:56:28

97

Views

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

Published on 23 Feb 2026 - 4:56

score 278 · Answer 3 · 2026-02-23 04:55:27

278

Views

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

Published on 23 Feb 2026 - 4:55

score 36 · Answer 4 · 2026-02-23 04:58:20

36

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 23 Feb 2026 - 4:58

score 368 · Answer 5 · 2026-04-01 10:16:46

368

Views

Box topology and connectedness property

Published on 01 Apr 2026 - 10:16

score 539 · Answer 6 · 2026-02-23 04:57:25

539

Views

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

Published on 23 Feb 2026 - 4:57

score 140 · Answer 7 · 2026-02-23 04:55:16

140

Views

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

Published on 23 Feb 2026 - 4:55

score 203 · Answer 8 · 2026-04-01 05:41:26

203

Views

Connectedness in box topology

Published on 01 Apr 2026 - 5:41

score 96 · Answer 9 · 2026-02-23 04:56:32

96

Views

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

Published on 23 Feb 2026 - 4:56

score 224 · Answer 10 · 2026-02-23 04:55:24

224

Views

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

Published on 23 Feb 2026 - 4:55

score 102 · Answer 11 · 2026-02-23 04:58:24

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 97 · Answer 12 · 2026-02-23 04:56:26

97

Views

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

Published on 23 Feb 2026 - 4:56

score 331 · Answer 13 · 2026-02-23 04:58:23

331

Views

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

Published on 23 Feb 2026 - 4:58

score 71 · Answer 14 · 2026-02-23 04:59:34

71

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 23 Feb 2026 - 4:59

score 69 · Answer 15 · 2026-02-23 04:53:25

69

Views

Difficulty in showing continuity of a function in different topologies

Published on 23 Feb 2026 - 4:53

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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