# Questions tagged [box-topology]

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16 questions
0answers
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### Is $\mathbb R^J$ normal in the box topology when $J$ is uncountable?

Question: Is $\mathbb R^J$ normal in the box topology when $J$ is uncountable? I know $\mathbb R^J$ is not normal in the product topology, see "Proof" that $\mathbb{R}^J$ is not normal ...
2answers
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1answer
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### Showing the $\ell^2$ topology is different from the Box topology

Let $X$ be the subset of $\Bbb{R}^\omega$ all of sequences $y = (y_i)$ for which $\sum y_i^2 < \infty$. Then $d(x,y) = \left[ \sum_{i=1}^\infty (x_i-y_i)^2 \right]^{1/2}$ defines a metric on $X$, ...
0answers
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### Prove that $\mathbb R_l$ $\times$ $\mathbb R_l$ given the box topology has a countable dense subset.

Let $\mathbb R_l$ be $\mathbb R$ given the lower limit topology, i.e. the topology generated by $\{[a, b)\subseteq \mathbb R|a<b\}$. Prove that $\mathbb R_l$$\times$$\mathbb R_l$ given the box ...
1answer
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### Projection map for the product, uniform and box topologies

That the canonical projection map (from say $\mathbb{R}^\mathbb{N}$ to $\mathbb{R}$) is continuous for all three topologies is straightforward. But is it also open for the uniform and box topologies? (...
1answer
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### Which Maps are Box-Continuous?

The box topology on the set $\mathbb R^\infty$ is defined to have sub-basis of sets $U_1 \times U_2 \times \ldots$ for each $U_i \subset \mathbb R$ open. Observe this is different from the product ...
5answers
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### Why are box topology and product topology different on infinite products of topological spaces?

Why are box topology and product topology different on infinite products of topological spaces ? I'm reading Munkres's topology. He mentioned that fact but I can't see why it's true that they are ...
1answer
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### What are the components and path components of $\mathbb{R}^{\omega}$ in the product, uniform, and box topologies?

I am working on an exercise problem about components and path components of $\mathbb{R}^{\omega}$. Specifically, Exercise about components and path components: 1. What are the components and path ...
5answers
3k views

### Is the box topology good for anything?

In point-set topology, one always learns about the box topology: the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where \$U_i \...