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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6 votes
1 answer
60 views

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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1 vote
3 answers
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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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2 answers
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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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  • 329
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1 answer
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Show that a set of sequences is open and closed on box topology

I have the following question on topology: Let $\mathbb R^\mathbb N = \prod_{n=0}^{\infty} \mathbb R$ with the box topology.Show that for all $\bar x = \langle x_n \rangle_n$, the set $ A = \{\bar y ...
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0 votes
1 answer
77 views

Connectedness in box topology

Let $X_i, \; i\in I$ be connected topological spaces and let $X=\Pi_{i\in I}X_i.$ I am familiar with the theorems that $X$ is a connected space in the product topology but not in the box topology. ...
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Let $[0, 1]^{\mathbb N}$ be equipped with the box topology.

Let $X = [0, 1]^{\mathbb N}$ be equipped with the box topology. Prove or Disprove that $[0, 1]^{\mathbb N}$ is connected. From what I've known $[0, 1]^{\mathbb N}$ is not compact based on this answer. ...
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1 vote
1 answer
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Countable product of $\mathbb{R}$ is not connected with respect to box topology

From the book Topology, by James Munkres, I am trying to understand that $\mathbb{R}^\omega$ is not connected with respect to box topology. It is written that, We can write $\mathbb{R}^\omega$ as the ...
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3 votes
1 answer
198 views

Box topology and connectedness property

I've been trying to properly comprehend how the box topology is constructed, which I noticed I don't understand corretly when my professor asked me to show if it's false or true that the box-topology ...
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1 answer
148 views

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

Here is my attempt at proving this, any ideas if it is completely correct? Let $U^{(0)}=[0,\frac{3}{5}), U^{(1)}=(\frac{2}{5},1]$. Let $x=(x_n)_{n \in \mathbb{N}}$ be an arbitrary sequence of $0's$ ...
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0 votes
1 answer
78 views

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

Find, for every $n \in \omega$, a non-constant function $f_n : \mathbb{R} \rightarrow \mathbb{R}$ so that $f: \mathbb{R} \rightarrow \mathbb{R}^{\omega}$, defined as $f(x) = (f_n(x))_{n \in \omega}$, ...
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4 votes
1 answer
85 views

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

The proofs I know for the fact that the space $[0,1]^\mathbb{R}$ is not first countable, use the product topology in some step of the demonstration. (Reference) So, I would like to know if this space ...
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2 answers
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Why is the function $f(t)=(t,t,t,t,t,...) $continuous in the product topology but not the box topology?

The function $f:\mathbb R \to \mathbb R^\infty$ given by $f(t)=(t,t,t,t,t\dots)$ is said to be not continuous in the box topology but continuous in the product topology. The example given is $U=(-1,1)\...
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1 answer
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Proving Theorem 19.2 in <Topology> by Munkres

Theorem 19.2 in Munkres Suppose the topology on each space $X_{\alpha}$ is given by a basis $\mathcal{B}_{\alpha}$. The collection of all sets of the form $$\prod_{\alpha \in J} B_{\alpha}$$ where $B_{...
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  • 197
2 votes
1 answer
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All the product topologies

I wonder what are all the topologies on the product of spaces that satisfy certain properties. For example, let $(X_i,\mathcal{T}_i)_{i\in I}$ be a family of topological spaces and let $X:=\prod_{i\in ...
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5 votes
1 answer
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Is the weak topology on $\mathbb{R}^{\infty}$ the same as the box topology?

Let $\mathbb{R}^{\infty}=\bigcup\limits_{n=1}^{\infty}\mathbb{R}^{n}$ be the subset of $\mathbb{R}^{\omega}$ consisting of all sequences which are nonzero for only finitely many terms. Give $\mathbb{...
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1 vote
2 answers
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Showing the infinite product space of $[0, 1]^\omega$ is not compact with respect to the box topology using the notion of open covers

I wish to show that the space $[0,1]^\omega$ in the box topology is not compact using the notion of open covers. The box topology has been a topic where I have struggled in my studies of topology, and ...
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Showing this set is open in the topological space $\mathbb{R}\times\mathbb{R}$

Let $\pi_1 : \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ be the projection on the first coordinate. Let $A$ be the subspace of $\mathbb{R}\times \mathbb{R}$ consisting of all points $(x,y)$ for which ...
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1 vote
1 answer
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$\mathbb R^{\omega}$ in the box topology not paracompact?

I know that we can't prove that $\mathbb R^{\omega}$ in the box topology is normal, so we can't say for sure that it is paracompact(it is Hausdorff, and every paracompact Hausdorff space is normal). ...
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Show that the $\mathbb{R^N} $ with the box topology is a regular space.

What was I thinking There is a neighborhood W=$\prod_i (-r_i,r_i)$ that does not intersect A . Consider a linear transformation $h_i:x_i \to x_i/r_i$ (linear transformation is a homeomorphism in both ...
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  • 603
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1 answer
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Box topology first countability

I want to show that $\mathbb{R}^2$ with the topology $\tau_\mathcal{B}$ generated by the base $\mathcal{B} = \{U \times V | U \in \tau_1, V \in \tau_2\}$ where $\tau_1$ and $\tau_2$ are topologies for ...
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1 vote
0 answers
168 views

Is $R^\omega$ in the box topology locally connected or locally path connected?

I have the following lemma: Give $R^\omega$ the box topology. Then $x$ and $y$ lie in the same component of $R^\omega$ if and only if the sequence $x-y$ is "eventually zero". Let $V_n=\{y\in ...
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0 votes
1 answer
75 views

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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  • 41
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1 answer
211 views

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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1 answer
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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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  • 131
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2 answers
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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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3 votes
2 answers
357 views

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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  • 951
2 votes
1 answer
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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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2 votes
1 answer
328 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 ...
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5 votes
1 answer
218 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 ...
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2 votes
2 answers
245 views

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 $...
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25 votes
2 answers
843 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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1 vote
1 answer
100 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 ...
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2 votes
1 answer
248 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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1 vote
2 answers
31 views

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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1 answer
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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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1 vote
1 answer
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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: ...
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4 votes
1 answer
112 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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3 votes
1 answer
396 views

Conditions to make box topology and product topology different

The product of the $A_j$ is the set $\prod_{j\in J} A_j$:=$\{ f: J \rightarrow \bigcup A_j \ | f(j) \in A_j \ for \ each \ j \in J \}$ We usually represent $f$ as $(a_j)=f(j) \in A_j$. For each $k \...
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1 vote
1 answer
135 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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  • 7,072
0 votes
0 answers
59 views

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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4 votes
1 answer
744 views

Continuity on Box Topology

I was reading about box topology and product topology from Munkres's Topology. Their is an example given below: $$\mathbb{R}^{\omega}=\prod_{n\in \mathbb{Z}_+}X_n$$ where $X_n=\mathbb{R}$ for each $n$....
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  • 5,163
2 votes
1 answer
156 views

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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  • 327
4 votes
1 answer
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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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  • 8,551
46 votes
6 answers
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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 ...
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  • 8,628
3 votes
1 answer
476 views

Understanding an open cover of $\prod_{n\in \mathbb{N}}[0,1]$ in the box topology

I am trying to understand why the following space $$X=\prod_{n\in \mathbb{N}}[0,1]=[0,1]^{\mathbb{N}}$$ is not compact with respect to the box topology. Specifically, I would like to understand a ...
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  • 243
17 votes
1 answer
6k 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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4 votes
2 answers
718 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 ...
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  • 5,825
3 votes
2 answers
750 views

Box topology on $\prod_{n=1}^\infty\mathbb{R}$

Let $X$ denote $\prod_{n=1}^\infty\mathbb{R}$, the Cartesian product of countably infinitely many copies of $\mathbb R$ (which is just the set of all infinite sequences of real numbers), endowed with ...
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