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.
599
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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 ...
2
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51
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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
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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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1
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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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1
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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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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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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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Supremum in $[-1, 1]^\omega$
The Problem
In this answer, a procedure for showing that every closed subspace of $[-1,1]^\omega$ is separable is given by making use of the lexicographic order. As far as I can tell, there are ...
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1
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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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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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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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Prove that $\begin{pmatrix} \mathbb{Q} \times \mathbb{R} \end{pmatrix} \cup \begin{pmatrix} \mathbb{R} \times \{0\} \end{pmatrix}$ is connected [duplicate]
According to a proposition on my class' notes, if $\{M_\lambda,\lambda \in \Lambda\}$ is a family of connected sets and $\bigcap\limits_{\lambda\in\Lambda}M_\lambda\neq\emptyset$, then $\bigcup\...
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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 ...
user733335
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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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0
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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 ...
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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{...
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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 ...
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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 ...
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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 $\...
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2
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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 $...
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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$ ...
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Connectedness of the set of continuous bijections $\Bbb R^n \to \Bbb R^n$
Consider $X = \{ f :\Bbb R^n \to \Bbb R^n \mid f \text{ continuous bijection } \} $ endowed with the product topology inherited possibly from $\Bbb R^{{n}^{\Bbb R^n}}$ (terrible notation). Is $X$ ...
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2
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Characterizations of product topology and box topology
Given a collection of topological spaces $X_i$ indexed by the elements $i$ of a set $I$, we consider the set product $P = \prod_{i \in I} X_i$ with projections $p_i : P \to X_i$. There are two methods ...
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Let $X = (-1,1)^{\Bbb N}$ have the product topology. Is the subset $(0,1)^{\Bbb N}$ open?
Let $X = (-1,1)^{\Bbb N}$ have the product topology. Is the subset $(0,1)^{\Bbb N}$ open?
To consider whether $(0,1)^{\Bbb N}$ is open I know that it is if I can find a basic open set that is a ...
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$\mathbb{R}_{\rm Sorg.}$ is paracompact but $\mathbb{R}_{\rm Sorg.} \times \mathbb{R}_{\rm Sorg.}$ is not
I’m trying to solve this problem: $\def\bbR{\mathbb{R}} \def\RSorg{\bbR_{\rm Sorg.}} \def\calR{\mathcal{R}} \def\calB{\mathcal{B}} \def\calC{\mathcal{C}} \DeclareMathOperator{\range}{range}$
Prove ...
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When does $(X\times Y,\Vert\cdot\Vert)$ have the product topology?
Let $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ be normed vector spaces and let $\Vert\cdot\Vert$ be a norm on $X\times Y$ such that $\Vert(x,0)\Vert=\Vert x\Vert_X$ and $\Vert(0,y)\Vert=\Vert ...
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Difficulty in showing continuity of a function in different topologies
Suppose that $f: \mathbb R\to \mathbb R^\omega$, where $\omega$ is countably infinite index set, is defined by $f(t)= (t,2t,3t,...)$.
The question is to discuss continuity of $f$ in the following ...
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1
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Measurability of diagonal set originated from Reflection Principle of Brownian motion
Let $B = (B_t)_{t\geq 0}$ be a Brownian motion from probability space $(\Omega, \mathcal{F}, P)$ to $(C(\mathbb{R}^+, \mathbb{R}), \mathcal{C}$), ($\mathcal{C}$ is the usual Borel $\sigma$-algebra ...
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Dense Compact subset of $X = ((\beta(D) \times (\mathfrak{c}^+ + 1)) \setminus (\beta(D) \setminus D) \times \{\mathfrak{c}^+\})$
Let $D$ be a discrete space of cardinality $\mathfrak{c}$ and $\beta(D)$ is the Stone Cech compactification of $D$. Let
$$
X = ((\beta(D) \times (\mathfrak{c}^+ + 1)) \setminus (\beta(D) \setminus D) \...
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3
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Continuity of minimum function on product spaces
Let $X, Y$ be topological spaces, and $Y$ is compact, and let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous function. Define $g: X \rightarrow \mathbb{R}$ as $g(x) = \inf_{y \in Y} f(x,y)$. ...
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Product of non-empty proper subsets is never open in an infinite product topology
If $\{X_i\}_i\in I$ are topological spaces, $I$ is infinite, $U_i \subset X_i$ are proper open subsets, then $\prod U_i$ is not open in $\prod X_i$
This is a question from my H.W. assignment. I’ve ...
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169
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Cartesian Product of Two Compact Sets is Compact
I'm trying to prove that the Cartesian Product of two compact sets is also compact, without the tube-proof thing.
So the rough draft of my proof is:
"Let (x(k), y(k)) be a sequence in AxB, such ...
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2
answers
47
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Question about infinite product topology
The product topology, sometimes called the Tychonoff topology, on ${\textstyle \prod _{i\in I}X_{i}}$ is defined to be the coarsest topology (that is, the topology with the fewest open sets) for which ...
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Show that a topological space $X$ is compact if and only if $X^\Bbb N$ is locally compact.
Show that a topological space $X$ is compact if and only if $X^\Bbb N$ is locally compact.
Assume that $X$ is compact, then for every open cover of $X$ there exists a finite subcover. To show that $X^...
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1
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107
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Comparing relative product topology, box topology and a final topology
Let $\Bbb R^\infty$ be the subset of $\Bbb R^\omega$ (the countably infinite product of $\Bbb R$ with itself) consisting of all sequences that are "eventually zero" that is, all sequences $(...
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1
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35
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Showing that $\bigcap_{i \in K}pr_i^{-1}(V_i) = \prod_{i \in I }V_i$
Let $X = \prod_{i\in I}X_i$ be a product space with the product topology. Then $X$ has a basis consisting of elements of the form $U = \prod_{i \in I} V_i$ where $V_i \ne X_i$ for only finitely many $...
- 1,394
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0
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35
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Associativity (?) of quotients
I have a relatively elementary topology question involving quotients of multiple spaces. I have two equivalence relations, $\sim_1$ and $\sim_2$. In my particular case, they have special meanings, but ...
- 158
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1
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37
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A proof on the connectedness of the product space of some connected spaces
I am trying to show that the product space of some connected spaces is also connected, and I am not sure whether I am right or not. Here is my proof:
Proof : Suppose that $\left\{X_\lambda\right\}_{\...
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0
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63
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Limit Point of a Subset using Open Ball
I am studying analysis II and I have come across the following lemma:
Lemma: Let $(V, ||.||)$ be a normed space, $E$ any subset of V. Then a point $\vec{y} \in V$ is a limit point of $E$ iff $(B_{r}(\...
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1
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Are all minimal closed sets boxes?
I'll set up a couple of (standard) definitions, then ask my question.
Definition: If $\mathbb X$ and $\mathbb Y$ are sets and $X\subseteq \mathbb X$ and $Y\subseteq \mathbb Y$, call $X{\times}Y=\{(x,y)...
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1
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Given the subbasis $\mathcal{S}_i$ of $X_i$, how may we construct a subbasis of $\prod_i X_i$ (in the product topology).
Let $(X_i,\tau_i)$ be topological spaces and the product topology $(X,\tau)$ we may construct, given the bases $\mathcal{B}_i$ of $\tau_i$, a basis $\mathcal{B}$ of $\tau$ as follows: any element $...
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