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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Restricted product topology and subspace topology on the ring of Adeles, which is finer?

Let $V := \{ \infty \} \cup \{ p \text{ prime} \}$, we define \begin{equation*} \mathbb{A} := \{ (x_v)_{v \in V} \in \prod_{v \in V} \mathbb{Q}_v \text{ with } x_p \in \mathbb{Z}_p \text{ for ...
The Tralfamadorian's user avatar
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Convenience formula for integral on product space

Let $\Omega_1 \times \Omega_2$ equipped with some product $\sigma$-algebra be a product space. Suppose $\mu$ is any positive measure (not necessarily any product measure) on $\Omega_1 \times \Omega_2$ ...
温泽海's user avatar
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Product topology and intersection of basis elements

I was wondering if you could help me settle a discussion with a co-author. Neither of us is a hard-core topologist but our current paper forced us into this, so we figured this forum was a good place ...
Paul_S's user avatar
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What is the lowest dimension for topological or isometric embedding of Y^n into euclidean space?

Let $Y$ denote the one-point union of three unit intervals. Since $Y$ embeds isometrically in the plane, it follows that the nth cartesian power $Y^n$ embeds isometrically — and also topologically — ...
Dan Asimov's user avatar
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Is the quotient map $q:X \times I \to CX$, where $CX$ is the cone of $X$, is open or closed?

Suppose $CX$ is the cone of $X$ defined by $CX := X \times I/ X \times \{0\}$. My question is the corresponding quotient map $q:X \times I \to CX$ is open or closed. Intuitively, it seems this should ...
Ramandeep Singh Arora's user avatar
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1 answer
80 views

Cell Structure of $S^2 \times S^2$, and more generally applying theorem to product of CW-complexes

My end goal is calculating the homology of $S^2 \times S^2$, comparing it to that of $\mathbb C P^2 \ \# \ \mathbb C P^2$, and using the cup product to show homology is insufficient as a comparison. I'...
George's user avatar
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Proving that the following basis generates a topology on the product space

I'm working through John M. Lee's Introduction to Topological Manifolds and am trying to complete exercise 3.25. In this exercise, we are asked to show that if $X_1, \dots, X_n$ are arbitrary ...
Keshav Balwant Deoskar's user avatar
1 vote
1 answer
67 views

Closed Set in Product Topology

I am trying to understand the Compactness Argument in a Graph Theory Problem using Probabilistic Methods. $V$ is infinite set. For each finite subset $X \subset V$, let $C_X \subset [2]^V$ be the ...
tom_choudhurry's user avatar
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1 answer
61 views

Equivalence of two metrics on countable product

There is an exercise 9.3#1 from "Topology without tears": Let $(X_i, d_i), i\in \mathbb{N}$, countable infinity of metric spaces, where every metric is bounded: $\forall X_i\forall a, b \in ...
Bogdan Shevchenko's user avatar
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proving $\mathbb{Q}^2$ is regular space in separately open topology.

I am studying the proof of theorem 5.5, regularity of the topology of separate continuity We are familiar with Euclidean topology on $\mathbb{R}^2$, in which open sets are defined with respect to the ...
Ashutosh Shinde's user avatar
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norm on product of normed linear spaces which induces the product topology

A norm in a linear space induces the topology defined by the metric \begin{equation} d(x,y)=\|x-y\| \end{equation} Consider a product of normed linear spaces $\Pi_{j\in J} V_j$. One can define the ...
liyiontheway's user avatar
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106 views

How to prove that $ \left \{ \prod_{i=1}^{n} U_i: U_i \text{ are open in } X_i \right \} $ is a base for the product topology?

I'm trying to prove that $$ B = \left\{ \prod_{i=1}^{n} U_i : U_i \text{ are open in } X_i \right\} $$ is a basis of the product topology. I was trying to use the subbase of the product topology, that ...
John's user avatar
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Is product of locally compact spaces always locally compact?

Suppose we have $X_1, \dots , X_n$ be locally compact topological spaces. We can Show that $X_1 \times \dots \times X_n$ is locally compact. By taking product of respective Compact sets which ...
Ezed's user avatar
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(in)equivalence of norms on product space

I'm looking for some light on this problem Let $A$ and $B$ be subspaces of a vector space $V$ such that $V=A\oplus B$, and let $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ be two norms on $V$. If $...
S. Chitratta's user avatar
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When does the Borel $\sigma$-algebra of compact convergence coincide with the product $\sigma$-algebra?

Let $X$, $Y$ be topological spaces, and $C(X,Y)$ the set of continuous functions $ X \to Y $, equipped with the compact-open topology. Let $\newcommand\Bco{\mathcal B_{\textrm{c-o}}} \Bco$ be the ...
Olius's user avatar
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Topology on the tangent space TM: Is this really the initial topology?

In a lecture series, I have come across the statement that the topology on the tangent space $TM$ is given by the coarsest topology which makes the projection map $\pi: TM \mapsto M$ continuous. In ...
P.Jo's user avatar
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2 votes
1 answer
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Right-triangle distance inequality in a product space

Let $(C_i)_{i=1}^m$ be nonempty closed convex subsets of a real Hilbert space $\mathcal{H}$. I am interested in proving (or finding a counterexample to) the following conjecture on the $m$-fold ...
Zim's user avatar
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1 answer
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Proof that a mapping into the product space is a topological embedding

I am self-studying Lee's "Introduction to Topological Manifolds" and am learning about the product topology in Chapter 3. There is a large proposition containing several properties of the ...
stuz's user avatar
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2 answers
200 views

How to prove $\mathbb Z^2 \ast \mathbb Z$ is quasi isometric to $\mathbb Z^2 \ast \mathbb Z^2$?

How to prove $\mathbb Z^2 \ast \mathbb Z$ is quasi isometric to $\mathbb Z^2 \ast \mathbb Z^2$? Here $\ast$ is the free product of groups. I am thinking of proving they are commensurable. In other ...
V SMASH's user avatar
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Prove explicitly that $\mathbb{R}^2_\text{usual} = (\mathbb{R}_\text{usual})^2$

The title says the question. Here's my proof. Let $\mathcal{B}_1$ be the basis for $(\mathbb{R}_\text{usual})^2$ and $\mathcal{B}_2$ be the basis for $\mathbb{R}^2_\text{usual}$. The elements of $\...
Ryukendo Dey's user avatar
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Show that $\operatorname{int}{(A \times B)} = \operatorname{int}(A) \times \operatorname{int}(B)$

The question goes as follows: Let $(X, \mathcal{T})$ and $(Y, \mathcal{U})$ be topological spaces. Let $A$ and $B$ be subsets of $X$ and $Y$ respectively. Show that $\operatorname{int} (A \times B) = ...
Ryukendo Dey's user avatar
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Under what conditions does [for any $x\in X$, $F(x,-): Y\to Z$ is continuous] imply [$F(x,y): X\times Y \to Z$ is continuous]?

Let $X,Y,Z$ be topological spaces and $F(x,y): X\times Y \to Z$ be mapping. Under what conditions does [for any $x\in X$, $F(x,-): Y\to Z$ is continuous] imply [$F(x,y): X\times Y \to Z$ is continuous]...
khkh's user avatar
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Property of joining of measure preserving systems (or of a measure on a product space).

Given two measure preserving systems $(X,\mathcal{B},\mu,T)$ and $(Y,\mathcal{C},\nu,S)$, a joining $\rho$ of those is defined as a $ T \times S$-invariant measure on the product $\sigma$-algebra, ...
User's user avatar
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$X\times Y$ and the product topology [duplicate]

I'm studing Andreas Gathmann's notes on algebraic geometry (pdf here: https://agag-gathmann.math.rptu.de/de/alggeom.php). In chapter 4 (about Morphisms) he was using the universal property of products ...
Schrödinger's cat's user avatar
5 votes
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Naive question about product topology vs quotient topology

I have a very naive question about the product topology. Whenever I see a topological space with a product topology structure, this structure is already explicitly given. That is, the statement will ...
GeorgeKenworthy's user avatar
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61 views

$X$ is compactly generated (not necessarily Hausdorff) and $Y$ is locally compact, then is $X\times Y$ still compactly generated?

While looking at a proposition from Hatcher AT: Proposition A.15. If $X$ is a compactly generated Hausdorff space and $Y$ is locally compact, then the product topology on X×Y is compactly generated. ...
Cezar's user avatar
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3 answers
106 views

For a closed subset $F ⊂ X × Y$ , the image $π(F )$ need not be closed in Y

I have the following question from an exam If $(X, d_X)$ is compact, show that every sequence in $X$ has a subsequence converging to a point of $X$. Deduce that the projection map $\pi$ then has the ...
Maths Wizzard's user avatar
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If the product of closed sets is closed in the product topology shouldn't {(0,0)} be closed in $A^1 \times A^1$ zariski topology.

I am not sure what is incorrect about the statement. If the product of closed sets is closed in the product topology shouldn't {(0,0)} be closed in $A^1 \times A^1$ zariski topology, i.e singletons ...
ben huni's user avatar
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1 vote
1 answer
41 views

Weak convergence on separable and complete product space

I read a paper in which the authors seem to have a simplified definition of convergence in distribution of random variables in a product space. The paper itself is very specific, so I can link it but ...
Tfid_dbg's user avatar
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78 views

Homology of the torus

I am coming from this post and i am fine with everything that happens there. I just want a few more details on the calculation of the effect of $d_2$. Following the way i learnt it we have to ...
Adronic's user avatar
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1 vote
1 answer
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Continuity of functions with multiple variables

I have started working on topology and there has been a question I can't figure out. Consider a function $f$ defined on topological spaces $X\times Y$ with values in $\mathbb{R}$. Both space are ...
chaki chaki's user avatar
1 vote
1 answer
155 views

Product topology and subspace topology

This seems to be a basic question but I can't get a proof. Suppose $X\times Y$ is the product space of non-empty topological spaces $(X, \tau)$ and $(Y, \rho)$. Let $\tau'$ and $\rho'$ be the ...
user760's user avatar
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4 votes
2 answers
363 views

Linear projections and topological direct sum in normed/Banach spaces

When learning about the concept of complemented subspaces of a Banach space, I'm curious with the following question: Let $X$ be a vector space over $\mathbb{C}$ or $\mathbb{R}$ and let $E$ and $F$ ...
user760's user avatar
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4 votes
1 answer
179 views

Let $X$ be a topological space such that $X \times \mathbb{R}$ is homeomorphic to $\mathbb{R}^2$. Must $X$ be homeomorphic to $\mathbb{R}$?

This question was posted on twitter here as a quiz but the author never gave an answer, so I thought I'd try here. I don't have much experience with topology so I'm stumped. From searching online it ...
Raphael's user avatar
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-1 votes
1 answer
95 views

Product topology and relativization [duplicate]

$(X_1,\tau_1)$ and $(X_2.\tau_2)$ are topological spaces with $Y_1\subset X_1; Y_2\subset X_2$. Let $X_1\times X_2=X; Y_1\times Y_2=Y $. Prove that the product topology on $Y$ obtained from topologies ...
111's user avatar
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2 votes
1 answer
60 views

Existance of product measure contradiction

I heard that given family of probability spaces $(\Omega_{\alpha}, \mathcal{F}_{\alpha}, \mu_{\alpha})_{\alpha \in A}$, there exist product measure $\mu$ on product sigma-algebra (smallest sigma-...
Esgeriath's user avatar
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1 vote
1 answer
57 views

Inclusion of continuous functions with compact open topology into product topology is continuous

Let $X,Y$ be topological spaces and $C(X,Y)$ the set of continuous functions from $X$ to $Y$ equipped with the compact open topology. It has a subbase consisting of sets $$V(K,U):=\{f\in C(X,Y)\ |\ f(...
user408858's user avatar
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2 votes
1 answer
48 views

Let $X = \{a,b,c\}$. Find a measure space $(X × X × X, \mathcal{A}, \mu)$ so that $\int_{X × X × X} \,d\mu =1$.

Let $X = \{a,b,c\}$. Find a measure space $(X × X × X, \mathcal{A}, \mu)$ so that $\int_{X × X × X} \,d\mu =1$. This question has me somewhat stumped because I'm not sure how to approach integrating ...
hzm's user avatar
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0 votes
1 answer
162 views

Product of $\sigma$-algebras Axler MIRA 5A.1

This question comes from reading Sheldon Axler's Measure, Integration and Real Analysis, and in particular form 5A.1. Let $\left(X,\,\mathcal{S}\right)$ and $\left(Y,\,\mathcal{T}\right)$ be ...
Cryo's user avatar
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1 vote
1 answer
108 views

The product and metric topologies on $X = \{1, 2, \ldots, n\}^{\Bbb Z}$ coincide

Consider $X = \{1, 2, \ldots, n\}^{\Bbb Z}$. We can endow $X$ with a topology in at least two ways: Let $\tau_1$ be the product topology on $X$, due to the discrete topology on $\{1, 2, \ldots, n\}$. ...
stoic-santiago's user avatar
4 votes
0 answers
60 views

Every sequential space is compactly generated, and both categories of spaces are cartesian closed. Do the products coincide for sequential spaces?

The categories of sequential spaces and compactly generated spaces both use a finer product than the one from $\textbf{Top}$ in order to be cartesian closed, and in both cases the product is arguably ...
saolof's user avatar
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0 votes
1 answer
48 views

A type of $T_2$ topological space such that for every point there's a $T_3$ subspace then the whole space is $T_3$ [closed]

Given a $T_2$ topological space $(X,\tau)$ such that $\forall x\in X, \exists V\in \tau$ such that $x \in V$ and the subspace $(cl(V),\tau_{cl(V)})$ is $T_3$ then $(X,\tau)$ is $T_3$. I really don't ...
user avatar
2 votes
1 answer
111 views

Class of cylinder sets and sigma algebra

I have the following definition for the class of cylinder sets $\mathcal{Z_J}$, with base $J\subset I $, where $I$ is an index set: $\mathcal{Z_J}=\{X_J^{-1}(A)\subset\Omega: A \in \mathcal{A_J}\}$ ...
Enrico's user avatar
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1 vote
0 answers
30 views

How do you determine the function representing the surjective map say for example $f: C\to S$ where $S=\left\{(x,y) \in R^2 : x^2+y^2=r^2\right\}\,?$

How do you determine the function representing the surjective map say for example $f: C\to S$ where $S=\left\{(x,y) \in\mathbb R^2 : x^2+y^2=r^2\right\}\;?$ $C$ in this question stands for the ...
Selena Krypton's user avatar
0 votes
1 answer
31 views

A property of a subset of the topological space $Y = \Pi_{n \in \mathbb{N}} X_n,$ where each $X_n$ is the Euclidean topology on $\mathbb{R}$

Let $X_n = \mathbb{R}$ for all $n \in \mathbb{N}.$ Now set $Y = \Pi_{n \in \mathbb{N}} X_n.$ Endow $Y$ with the product topology, where the topology on each $X_n$ is the Euclidean topology on $\mathbb{...
Hari Krishnan's user avatar
2 votes
1 answer
345 views

Defining an Embedding in the Product Topology $f: X \to X \times Y$ and $f(x) = x \times y_0$ for $y_0 \in Y$

Munkres defines a topological embedding as follows. Now suppose that $f:X\to Y$ is an injective continuous map,where $X$ and $Y$ are topological spaces. Let $Z$ be the image set $f(X)$, considered as ...
Talmsmen's user avatar
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$(X\times [0,1])/\sim$ homeomorphic to $X\times S^1$

The equivalence relation $\sim$ is given by $(x,0)\sim (x,1) \quad\forall x\in X$. I already know that $[0, 1]/ \{0, 1\}$ is homeomorphic to $S^1$ but have problems showing this for the given product ...
Lu1998's user avatar
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1 vote
1 answer
82 views

Which of the following subsets of $\mathbb{R}^{\omega}$ are open sets in product topology?

This question is likely to be trivial, but I'm confused by it. I will leave here a couple of the examples my teacher left here, so I can guide myself through the others. Which of these subsets of $\...
José Pedro Ferreira's user avatar
2 votes
2 answers
91 views

Is the factor of a topological product closed in the product topology?

I could not find an answer to the question of whether a factor of a topological product is closed in the product topology itself, so I wrote my own proof. My questions are under the proof itself. Let $...
Moguntius's user avatar
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2 votes
1 answer
50 views

Let $X = \{ f \in \Bbb Q^\Bbb R : \sum_{x \in \Bbb R} |f(x)| < \infty \}$. Show that no sequence in $X$ converges to the constant function $1$.

Let $X = \{ f \in \Bbb Q^\Bbb R : \sum_{x \in \Bbb R} |f(x)| < \infty \}$. Show that $X$ is a dense subset of $\mathbb{R}^{\mathbb{R}}$ with the product topology and that no sequence in $X$ ...
Kurosaki's user avatar

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