# 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 ...
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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 $\Omega_1 \times \Omega_2$ ...
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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 ...
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4 votes
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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 — ...
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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 ...
1 vote
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### 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'...
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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 ...
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1 answer
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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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4 votes
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### 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 ...
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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]...
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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, ...
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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 ...
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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 ...
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### $X$ 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. ...
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3 answers
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 $F$ ...
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4 votes
1 answer
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### 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 ...
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1 answer
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### 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 ...
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2 votes
1 answer
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### 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-...
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1 answer
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### 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(...
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2 votes
1 answer
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### 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 ...
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### 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 ...
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### 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 $\{1, 2, \ldots, n\}$. ...
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4 votes
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### 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 ...
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### 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 ...
2 votes
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### 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: $\mathcal{Z_J}=\{X_J^{-1}(A)\subset\Omega: A \in \mathcal{A_J}\}$ ...
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### 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 ...
0 votes
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