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Questions tagged [product-space]

For questions about the structure of product space, in the context of topology (including metric and normed spaces) or measure theory. Use other tags to indicate the context.

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11 views

Does a sequentially continuous function take its supremum on compacts?

Consider the following situation: Let $X$ be a separable metric space [if this helps: I am mainly interested in the case $X = \mathcal{P}(\mathbb{R}^d)$, the space of all Borel-probability measures on ...
4
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1answer
30 views

When is the product of closed sets closed in the product topology?

I have a specific example below and i think my proof is wrong because it seems too simple, it would work for the general case which I doubt is true. I would also be interested in the general answer ...
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2answers
69 views

$+:X\times X\to X,(x,y)\mapsto +(x,y)=x+y$ and $\cdot:\Bbb{R}\times X\to X,(\lambda,x)\mapsto \cdot(\lambda,y)=\lambda\cdot x$ are weakly continuous

$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed ...
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1answer
25 views

Cellularity of product space

Let $\{X_s\}_{s\in S}$ be a family of topological space, and $d(X_s)\leq m$, then the cellularity, i.e., the supremum of the cardinalities of all families of pairwise disjoint non-empty open subsets ...
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36 views

Every bounded net in a dual space has a cluster point?

Question: Let $X$ be a normed space and $X^*$ be its continuous dual of $X.$ Assume that $(x_\alpha^*)_\alpha$ is a bounded net in $X^*.$ Is it true that there exists a cluster point $x^*$ in $X^*$ ...
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1answer
37 views

Is the subspace topology of the product topology the initial topology with respect to the restrictions of the projection maps?

Let $I$ be an index set, $X_i$ be a topological space for each $i \in I$ and $X = \prod_{i \in I} X_i$ the product of all $X_i$. Then the product topology is exactly the initial topology with respect ...
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1answer
19 views

How do I prove the $f:X\to I$ by $f(x)=\min\{f_i(x_{\beta_i}):i=1,2,…,n\}$ is continuous.

How do I prove that $f:X\to I$ by $f(x)=\min\{f_i(x_{\beta}):i=1,2,...,n\}$ is continuous. My attempt:- Minimum of two continuous real-valued functions are continuous. $$f_i \circ \pi_{\beta_i}: X\...
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1answer
29 views

Is $D$ closed in $\Bbb R^n$? [closed]

Let $C$ be a non-empty closed subset of $\Bbb R^n$ and $C × D$ be a closed subset of $C × \Bbb R^n$. Can we say that $D$ is closed in $\Bbb R^n$?
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65 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
58 views

Product of a compact topological space and a singleton in another topological space is compact proof

It's before we prove that 'Product of two compact sets is compact'. There are topological spaces $X$(which is compact), $Y$ and the product topology on $X \times Y$ is given by the subbase $U \times V$...
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2answers
43 views

Product of a compact set and a singleton is compact proof

It's before we prove that 'Product of two compact sets is compact'. Let $S$ be an open cover of $X \times \{\bullet\}$ where $X$ is compact. Then $\pi_1(S)$ is an open cover of $X$ so there is a ...
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1answer
18 views

Proving there exists $V$ so that $\{x_0\}\times I\subset V\times I\subset U$

Let $X$ be a topological space, and $I=[0,1]$. Consider $X \times I$ with the product topology. Now fix $x_0\in X$ and $U\subset X\times I$ an open that contains $\{x_0\}\times I$. Prove that there ...
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2answers
60 views

Question regarding proof of Tychonoff's theorem

On Wikipedia it states that a space $X$ is compact if and only if every net has a convergent subnet. It then states that a net in the product topology has a limit if and only if each projection has a ...
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0answers
29 views

Is $[0,1]^A$ not sequentially compact for any uncountable $A$?

We know $\{0,1\}^\mathbb{N}$ is sequentially compact, but $\{0,1\}^\mathbb{R}$ is not. The proof I have seen that the second is not sequentially compact heavily relies on the fact that we have a ...
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0answers
15 views

Questions about Tychonoff spaces natural embedding.

On Wikipedia it states that for any Tychonoff space $X$ there is a natural embedding into $[0,1]^{C(X,[0,1])}$. I assume this embedding is $\iota(x)(f)=f(x)$. I am able to prove that $\iota$ is ...
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0answers
45 views

Do separately semi-continuous functions have a dense set of semi-continuities?

The connection between separate continuity and joined continuity has been studied quite a lot. In particular, one has (as a special case of a far more general Theorem from here) the following: If $...
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1answer
52 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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0answers
14 views

Differentiable structure on product of manifolds to yield inclusion maps as imbeddings

I am working through Munkres' "Elementary Differential Topology" and trying to do every exercise, but this one question has me somewhat stuck. It is exercise c on page 11. The exercise is as follows: ...
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1answer
43 views

Banach-Alaoglu theorem, Rudin's functional analysis.

Few questions about the theorem If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$ K = \left\{\lambda \in X^* : |\Lambda x | \leq 1 \; \text{for every} \; x \in V \right\}...
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1answer
52 views

Example of product topology where the index set is uncountable

I'm reading through Munkres, chapter 2, section 19 (Product topology). I can't see any example of product topology where the family of space is indexed with an uncountable set. Can you provide an ...
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2answers
56 views

What is {empty set} x something with the product topology?

The product topology is defined as the topology induced by the basis of the product of open sets from each of the original topologies. From what I understand, then $\{\emptyset\}x]a,b[$ is an open ...
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2answers
123 views

Can every locally compact Hausdorff space be recognized as a subspace of a cube that has an open underlying set?

In this question cubes are topological spaces of the form $[0,1]^J$ with product topology and $[0,1]$ with usual topology. Further a space is a Tychonoff space if and only if it is a completely ...
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2answers
40 views

Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Let X be a Banach space. Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology. Could anybody give me some hints to start?
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1answer
66 views

Help with product maps in topological spaces

Let $A, B, X, Y$ be topological spaces. Given two functions $f : A \to B$ and $g : X \to Y $, let $f \times g : A × X \to B \times Y$, $ (f \times g)(a, x) = (f(a), g(x))$. can you help me to show ...
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1answer
37 views

Set of continous functions writable as cartesian product

let $(X,d_X)$ and $(\mathbb{R}^n,d_2)$ be metric spaces with $X$ compact. Then the set of continuous functions is defined by $$ C_n(X):=\{f:X\rightarrow \mathbb{R}^n \;|\;f \text{ continuous }\} $$ $\...
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1answer
60 views

Topology closed sets in products of spaces

I got stuck at question 3.7.1 in Bert Mendelson's introduction to Topology. Prove that a subset $F$ of $X = \prod_{i=1}^nX_i$ is closed if and only if F is an intersection of sets, each of which is ...
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1answer
63 views

Formula for relative homotopy groups of products

In Hatcher prop 4.2 he proves that the n-th homotopy group of a product $X\times Y$ (for $X$ and $Y$ path-connected) is isomorphic to the product of the n-th homotopy groups of $X$ and $Y$. I wonder ...
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1answer
71 views

If $X \times X$ is normal, then is $X \times X \times X$ normal?

I am looking at some topological dimension theory for product spaces, and in trying to construct a certain type of counterexample it's become relevant to consider the question in the title above. I ...
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0answers
24 views

A product of intervals where the indexing set is all bounded sequences; Generalizing the limit of a sequence so that it is always defined.

This problem seems very hard to me, and any hints will be greatly appreciated. Here is the question: Let S be the space of bounded real sequences; for each $a\in S$ let $M_a$ be its least upper ...
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1answer
37 views

Compactness and countably compactness in metric spaces

I have a little doubt about compactness in metric spaces. I have this homework where I have to prove that $[0,1]^\omega$ with the uniform topology is not countably compact. As a consequence of ...
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1answer
40 views

Countability of Topologies and Product Topology

I was trying to examine examples of topologies as a self learning project (you can see the list here), and these questions arose: I couldn't spot a product topology where any set other than basis ...
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1answer
26 views

Non-homeomorphic structures and the Descartes' theorem

Some structures like the donut are not homeomorphic to a sphere. According to this link (https://en.wikipedia.org/wiki/Angular_defect#Positive_defects_on_non-convex_figures) the basis of the Descartes'...
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1answer
32 views

Continuous function mapping non-open sets to open sets

While studying the product topology, I've come across an example that I can't shake. Consider the product topology on $\mathbb{R}^2$. We know that 1) This product topology is the same as the "...
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1answer
68 views

Embedding of rational number Q into real number R is dense and the embedding of Q into p-adic numbers $Q_p$ is dense too

Q: If $p,q$ are different primes, show the embedding $$\mathbb{Q} \rightarrow \mathbb{Q}_q \times \mathbb{Q}_p$$ $$x \rightarrow (x,x)$$ is dense in the product space $\mathbb{Q}_q \times \mathbb{Q}_p$...
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2answers
22 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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1answer
27 views

Ultrafilters over product spaces

Suppose that for $i\in I$, $X_i$ are topological spaces and $U_i$ is an ultrafilter over $X_i$. Consider the space $\Pi_i X_i$ with the product topology. I want to know when, if ever, it is possible ...
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1answer
20 views

Elements in the basis of Product topology determined by sub-basis other than sub-basis elements.

I could prove the result for $|\Lambda|$ finite. Here $|\Lambda|$ is arbitrary. My attempt:- Let $\langle x_{\alpha}\rangle_{\alpha\in \Lambda}\in B \implies \langle x_{\alpha}\rangle_{\alpha\in \...
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1answer
25 views

When is a $(\prod_{\alpha \in J} X_\alpha \to X)$-function continuous?

I know that a function $X \to \prod_{\alpha \in J} X_\alpha$ from a topological space into a space with the product topology is continuous if and only if each component function $f_\alpha : X \to X_\...
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0answers
41 views

Compact subspaces of the sigma product

Let $\prod_{\alpha<omega_1}(I)$ be the Tychnoff product of the unit interval $I$, then the product topology is Hausdorff compact topology. We know that the sigma product $\Sigma(0)$ as a subspace ...
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1answer
68 views

What do opens in the product topology over $\mathbb{Z}_p$ look like?

Let $p$ be a prime, let $\pi_n : \mathbb{Z}/p^{n+1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$ be the obvious projection and $$\mathbb{Z}_p = \{\, x \in \Pi_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z} \,|\, \...
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1answer
68 views

Topology of $\mathbb{R}^{\infty}$ [duplicate]

I want to know which is the topology of $\mathbb{R}^{\infty}$, but I don't know even how to start to give it one. How can I give to $\mathbb{R}^{\infty}$ a topology? Is there a book that explains this?...
2
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2answers
56 views

Understanding the Product Topology

This is pretty basic, but I just learned about the product topology in class, and it seems to me that it doesn't agree with how it is defined - obviously, I'm wrong, but I was hoping someone could ...
2
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2answers
85 views

Alternative proof for subbasis for order topology and product topology: Finite intersections of elements of $\mathscr S$ is a basis

Topology by James Munkres: Both of the following proofs are at the very end of, respectively, Sections 14 and 15, which seem to correspond with the definition of a subbasis' being at the very end of ...
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1answer
61 views

Set which is locally $\iota-$null but not $\iota$-null

I've been working on this problem for a while but cannot seem to reach a solution. Let $\iota(A)= \bar{\bar{I}}(\xi_A)$ where $\xi_A$ is the characteristic function of the set $A=\{(x,0) | x \in \...
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1answer
36 views

How to show that if $U$ is an open set in a product topology then $\pi_{i}(U)=E_i$ for all but finitely many indices $I.$

Problem Statement: Let $((E_i,T_i))_{i\in I}$ be a family of non-empty topological spaces. If $U$ is any $\Pi_{i\in I} T_i$-open subset of $\Pi_{i\in I}E_{i}$ prove that $\pi_{i}(U)=E_i$ for all but ...
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1answer
58 views

Measurability of an uncountable union

Let $\{X_i\}_{I \in I}$ be a family of real-valued random variables on $(\Omega,\mathcal{A},P)$ for an uncountable index set $I$, which is the sample space of $(I,\mathcal{F},Q)$. On the product ...
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0answers
100 views

Separate continuity and analyticity in one variable implies joint continuity?

A theorem of Hartog states that if $U \subseteq \mathbb C$ is open and $f : U \times U \to \mathbb C$ is analytic when we fix any variable ('separately analytic'), then it is continuous. In general, ...
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2answers
353 views

If a function on a product space is continuous in each variable, is it locally bounded?

Let $f : \mathbb R \times \mathbb R \to \mathbb R$ be continuous when we fix one variable. Then $f$ need not be continuous (see e.g. Functions continuous in each variable ). Does it imply that $f$ ...
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1answer
50 views

accumulation points in $\mathbb R$

Consider the subset $A = [a, b)$ of $\mathbb R$. Then it is easily verified that every element in $[a, b)$ is a limit point of $A$. The point $b$ is also a limit point of $A$. Why?
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0answers
14 views

If $P_A(\theta^{-1}(V\times D)) = U\text{ and } P_B(\theta^{-1}(V\times D)) = B,$ can we conclude that $\theta^{-1}(V\times D) = U\times B$?

Let $A,B,C,D\subseteq \mathbb{R}$ be sets. Denote $\theta:A\times B\to C\times D$ a homeomorphism (bijective continuous with continuous inverse), where $$A\times B = \{(a,b): a\in A,b\in B\}\quad\...