# Questions tagged [box-topology]

The box topology is the topology on the cartesian product of sets generated by the cartesian product of open subsets in each component set. Use this tag when your question involves the box topology.

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

Let $X$ be a nontrivial topological space, $I$ be a infinite set, we can endow $X^I$ (the set of all functions $I\to X$) with either the product topology or the box topology. We know that the box ...
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### Where does this "proof" that $\Bbb{R}^\omega$ is normal in the box topology go awry?

James Munkres' Topology, 2nd Edition indicates that the space $$\Bbb{R}^\omega := \{ (x_0, x_1, x_2, ...) | x_i \in \Bbb{R}, \forall i < \omega \}$$ equipped with the box topology is completely ...
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### 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.

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. To show that $A$ is open I think I don't need to consider $A^c$. ...
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### $\prod_{n=1}^{\infty}{\mathbb{R}}$ endowed with the box topology is not first countable.

What I'm trying to prove is that if $X^{+}\subseteq X:=\prod_{n=1}^{\infty}{\mathbb{R}}$ is the set of all positive sequences in $\mathbb{R}$, then no sequence of elements in $X^{+}$ converges to the ...
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### Showing a subset is open with respect to a topology

I have asked and read similar questions, but I am still somewhat confused on the notion of "a set being open with respect to some topology". My task is concerning the box topology and the ...
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### Showing a set is open with respect to box vs product topology

I am currently studying topology with Munkres and I am asking for some general proof techniques and clarification rather than just posting my question and wait for a solution. I am given that the real ...
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### Is it possible to show $\mathbb{R}^\omega$ is not metrizable without using the first countable definition and sequence lemma?

I encountered a question about proving that $\mathbb{R}^\omega$, which is the countably infinite product of $\mathbb{R}$ under the product topology, is not metrizable. I have seen many solutions here ...
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### What's the interior of $R^{\infty}$ in the $R^{\omega}$ with the box topology and the production topology?

What's the interior of $R^{\infty}$ in the $R^{\omega}$ with the box topology and the production topology? With $R^{\infty}$ being the set of all eventually zero sequences. About the question in Prob. ...
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### Box topology and a non-continuous function

I'm having trouble understanding box topologies on infinite spaces. I'm asked to prove that the function f going from ([0,1], standard top.) to ([0,1]^N, box top.) is not continuous. This function ...
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### Net convergence over product and box topology

Let $(x_\alpha)_{\alpha\in \Lambda}$ a net in the topological space $X = \prod_{i\in I}X_i$. Is it true that $x_\alpha \rightarrow x$ iff $p_i(x_\alpha) \rightarrow p_i(x)$ for every $i \in I$ with ...
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### dimension-set for all Julia-fractals/Mandelbrot set

You know the Julia-sets have all different not integer dimensions. I think it would be very interesting to have a map like the Mandelbrot-set, that shows the magnitude of the dimension of the Julia-...
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### Which properties does the box topology conserve?

The Wikipedia article on the Product topology has a wealth of examples of properties conserved by the product topology. The following is a quote from the linked article: " Separation Every product ...
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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 ...