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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.

3
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2answers
31 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 ...
2
votes
0answers
22 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 ...
1
vote
0answers
14 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 ...
1
vote
1answer
50 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 ...
0
votes
0answers
12 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: ...
0
votes
1answer
36 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\}...
1
vote
1answer
49 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 ...
-1
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2answers
51 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 ...
3
votes
2answers
116 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 ...
0
votes
2answers
37 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
65 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 ...
1
vote
1answer
36 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 $$ C_n(X):=\{f:X\rightarrow \mathbb{R}^n \;|\;f \text{ continuous }\} $$ $\...
1
vote
1answer
58 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 ...
0
votes
1answer
61 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 ...
4
votes
1answer
66 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 ...
1
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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 ...
0
votes
1answer
36 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 ...
0
votes
1answer
33 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 ...
0
votes
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'...
2
votes
1answer
30 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 "...
1
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1answer
61 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 $\mathbb{Q}_q \times \mathbb{Q}_p$...
1
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2answers
21 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. ...
0
votes
1answer
25 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 ...
0
votes
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 \...
0
votes
1answer
24 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_\...
0
votes
0answers
39 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 ...
3
votes
1answer
62 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} \,|\, \...
0
votes
1answer
65 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
votes
2answers
46 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
votes
2answers
59 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 ...
1
vote
1answer
56 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 \...
1
vote
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 ...
0
votes
1answer
50 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 ...
2
votes
0answers
67 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, ...
6
votes
2answers
343 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 $f$ ...
-1
votes
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?
0
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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\...
4
votes
1answer
49 views

Measurability of piecewise defined function on a product space

Let $(\Omega,\mathcal A)$ and $(X,\mathcal X)$ be measurable spaces $g_n:\Omega\times X\to\mathbb R$ be $\mathcal A\otimes\mathcal X$-measurable for $n\in\mathbb N$ Assume that for all $x\in X$, ...
2
votes
1answer
27 views

Coproduct of totally disconnected sets is totally disconnected.

Let $(X_i)_{i \in I}$ be a family non empty topological spaces. Prove that $\coprod_{i \in I} X_i$ is totally disconnected if $X_i$ is totally disconnected for all $i \in I$ My book gives the hint: ...
2
votes
1answer
21 views

$\mathcal{S}:= \{\operatorname{pr}_k^{-1}(A_k) \mid k \in I, A_k \in \mathcal{A_k}\}$ subbasis for product topology.

Let $(X_i, \mathcal{T}_i)_{i \in I}$ be a family non empty topological spaces with basises $(\mathcal{A_i})_{i \in I}$ Prove that $\mathcal{S}:= \{\operatorname{pr}_k^{-1}(A_k) \mid k \in I, ...
1
vote
2answers
80 views

Uncountable product of second countable spaces

Let $(X_i, \mathcal{T}_i)_{i \in I}$ be a family non indiscrete topological spaces and equip the product with the product topology $\mathcal{T}$. If $\prod X_i$ is second countable, prove that $|I|...
7
votes
2answers
560 views

Meaning of “the weakest topology such that <blank> is continuous”

I've noticed one classical way of defining certain topologies is to define them as the "weakest" (or coarsest) topology such that a certain set of functions is continuous. For example, The product ...
4
votes
1answer
126 views

Is this statement really that technical to prove?

Let $\{X_i\}_{i\in I}$ a family of topological spaces and $X=\prod_{i\in I}X_i$ denote the product topology, i.e. a subbasis $\mathcal{C}$ of it is given by $$\mathcal{C}=\left\{\left.{\textstyle\...
-1
votes
2answers
36 views

Why are facts proven in the (infinite) product topology also true for finite products?

Context question: ZFC Let $(X_i, \mathcal{T}_i)_{i \in I}$ be a family non empty-topological spaces. We define the product space as the topological space $(\prod_{i \in I} X_i, \prod_{i \in I} \...
0
votes
1answer
38 views

Is $\prod_{i \in I} f_i$ continuous?

Let $I \neq \emptyset, (X_i \neq \emptyset, \mathcal{T}_i)_{i \in I}$ be a family topological spaces.. Choose (using AC) an element $(x_i)_{i \in I} \in \prod_{i \in I} X_i \neq \emptyset$ Define for ...
2
votes
3answers
39 views

Cartesian product indexed by a closed interval

Let $I=[0,1]$ and $$X = \prod_{i \in I}^{} \mathbb{R}$$ That is, an element of $X$ is a function $f:I→\mathbb{R}$. Prove that a sequence $\{f_n\}_n ⊆ X$ of real functions converges to some $f ∈ X$ in ...
1
vote
0answers
28 views

What is the value of $K$ in Talagrand's inequality?

In a paper by Talagrand, the following theorem is proved. Theorem 1.4. There exists a number $K$ with the following property. Consider $n$ independent random variables $X_i$ valued in a ...
1
vote
1answer
27 views

Connectedness, Products of Finite Topologies

I will show that for all $ i\in I$ (finite), $X_{i}$ is connected implies that $\prod_{i\in I} X_{i}$ is connected. I have already shown that $X_{1}$ and $X_{2}$ are connected. Now I am supposed to ...
-1
votes
2answers
42 views

Continuity of $f(x,y)=x$ in Product Metric [closed]

Let $(X \times Y,d)$ be the product metric space of $(X,d_1)$ and $(Y,d_2)$. Show that the function goes to $X$ from $X \times Y$ defined by $f(x,y)=x$ is continuous. I really have any ideas about ...
1
vote
1answer
111 views

Prove the metric induced topology is the same as the product topology

Let $P = \prod_{n=0}^\infty I_n , \ I_n = \left[0,1/n\right] $ and $d(\bar x,\bar y) = \left [ \sum_{n=0}^\infty (x_n-y_n)^2 \right]^{1/2} for\ \bar x,\bar y \in P.$ Prove that the topology induced by ...