# 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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• 267
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### 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. ...
• 3,416
1 vote
0 answers
135 views

### 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. ...
• 53
1 vote
1 answer
494 views

### 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 ...
• 305
3 votes
1 answer
356 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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### $X=\prod_{j\in J}X_j$ is Hausdorff(regular) iff each $X_j$ is Hausdorff(regular) on box-topology. Then what about the $T_1$ and product-topology?

The box-topology proof is quite obvious as $\prod{U_j}$ is always open in $X$, where the product-topology may not stand. By the way, if regularity stands, does this mean $T_1$ axiom stands first? Then,...
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1 vote
1 answer
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### 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 $0's$ ...
0 votes
1 answer
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### 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
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### 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 ...
• 2,767
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2 answers
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• 307
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1 answer
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• 1,876
1 vote
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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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0 votes
1 answer
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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
133 views

### $\mathbb R^{\omega}$ 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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0 votes
1 answer
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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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### 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 ...
1 vote
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• 6,523
29 votes
2 answers
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### 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
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### 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
360 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
69 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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### 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 ...
2 votes
1 answer
120 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: ...
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4 votes
1 answer
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• 459
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1 answer
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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 $X$, ...
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3 votes
0 answers
358 views

### Connected components of $\mathbb{R^w}$ in box topology [duplicate]

How can I know about all $\mathbb{R^w}$ in box topology? How can I separate $\mathbb{R^w}$ in box topology in to components and path components?in uniform topology I can separate using the path ...
0 votes
0 answers
63 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
1k 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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2 votes
1 answer
184 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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