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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questions
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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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2answers
27 views
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 ...
2
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1answer
51 views
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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1answer
104 views
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 ...
4
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1answer
138 views
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 ...
2
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2answers
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Nontrivial example of continuous function from $\mathbb R\to\mathbb R^{\omega}$ with box topology on $\mathbb R^{\omega}$
Nontrivial example of a continuous function from $\mathbb R\to \mathbb R^{\omega}$ with box topology on $\mathbb R^{\omega}$ and usual topology on the domain.
I can give some trivial example like
$...
14
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1answer
304 views
Does the box topology have a universal property?
Given a set of topological spaces $\{X_\alpha\}$, there are two main topologies we can give to the Cartesian product $\Pi_\alpha X_\alpha$: the product topology and the box topology. The product ...
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1answer
66 views
Proving $(-1,1)^{\mathbb{N}}$ is not open in the product topology of $\mathbb{R}^{\mathbb{N}}$
Clarification: here $\mathbb{R}^{\mathbb{N}} = \mathbb{R}\times \mathbb{R} \times \cdots$, i.e, countably many copies of $\mathbb{R}$. $(-1,1)^{\mathbb{N}}$ is completely analagous.
I don't want a ...
2
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1answer
148 views
Is $\mathbb{R}^\omega$ endowed with the box topology completely normal (or hereditarily normal)?
Just out of curiosity, I'd like to know more properties of box topology. I found Is $\mathbb{R}^\omega$ a completely normal space, in the box topology? quite interesting, but unfortunately, it hasn't ...
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2answers
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Is the sequence of real number that are $0$ from some point Sense in the box or product topologies
Let $A=\{(x_n)_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}|\exists M\in\mathbb{N} ,\forall n>M, x_n=0 \}\subset\mathbb{R}^\mathbb{N}$, series of real numbers that are zero from some point forward.
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1answer
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Closures and interiors of real sequences
Let X be the set of all real sequences. For both the box and the product topology, find the closure and the interior of S, where S is the subset of X containing:
(a) bounded sequences
(b) sequences ...
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1answer
39 views
Space of Sequences with Finitely Many Nonzero Terms is Paracompact
I just proved the following theorem:
If $X$ is a regular space that can be written as a countable union of compact subspaces of $X$, then $X$ is paracompact.
I am now working on the following:
...
4
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1answer
59 views
Box topology defines a topological vector space?
Is the set of all real summable sequences, endowed with the box topology, a topological vector space?
Formally, I am interested in $X=\{(a_1,\ldots):a_n \in \mathbb{R} \ \forall n, \ \sum a_n < \...
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1answer
75 views
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 $X$, ...
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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 ...
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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? (...
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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 ...
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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 ...
13
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1answer
4k views
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 ...
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2answers
492 views
Difference between the behavior of a sequence and a function in product and box topology on same set
Let $\prod_{\alpha \in J} X_{\alpha}$ is the product of typologies. Consider product topology on the set. Then any function $f : A \rightarrow \prod_{\alpha \in J} X_{\alpha}$ will be continuous on ...
13
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2answers
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Tychonoff Theorem in the box topology
A short question: Why does not the Tychonoff theorem (the arbitrary product of compact spaces is compact) hold in the box topology?
I don't know how to show that there is no finite sub-cover of any ...
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3answers
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$[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact
could anyone help to show that $[0,1]^{\mathbb{N}}$ with respect to the box topology is not compact? Thank you!
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5answers
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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 \...
