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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1answer
32 views

Distance to the boundary

Let $F$ be a closed set in $\mathbb{R}^n$ and $\Omega:=\mathbb{R}^n\setminus F$. For $x\in\Omega$, do we have ${\rm dist}(x,F)={\rm dist}(x,\partial \Omega)$?
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1answer
36 views

Showing a set is closed in the product topology

We have some topological space $X$ with continuous functions $f,g:X \rightarrow \mathbb{R}$ equipped with the usual topology I want to show that then set: $$E = \{(x,y):f(x)=g(y)\} \subset X \times X $...
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1answer
33 views

Product of a $k$-space and a compact space

I am beginning to learn about compactly generated spaces. I would like to know whether the following is true: if $X$ is a compact Hausdorff space and $Y$ is Hausdorff compactly generated space, then ...
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1answer
53 views

Open set in product topology which is not “cylindrical”

I know that the product topology on $\prod_{i=1}^\infty\mathbb{R}$ has a basis of the form: $$ \prod_{i=1}^N U_i \times \prod_{i=N+1}^\infty \mathbb{R} \tag{*} $$ where $U_i\subseteq \mathbb{R}$ is ...
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0answers
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How do you denote functions inherited by a product space?

Given a map $f:A\to B$, define the map $p_n(f):A^n\to B^n$ as $p_n(f)(a_1,...,a_i)=(f(a_1),...,f(a_i))$. Equivalently, you could say given $f$, $p_n(f)$ is such that $\mathrm{proj}_a\circ p_n(f)=f\...
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1answer
41 views

About the quotient space $Z/p^iZ$.

Fix a prime number $p$ and consider $Z_p=\{(a_i)_{i\in N}:a_i\in Z/p^iZ_p\ and\ \phi_{i+1}(a_{i+1})=a_i\ for\ every\ i \}\subset \prod_{i=1}^{\infty}(Z/p^iZ)$, where $\phi_i:Z/p^iZ\rightarrow Z/p^{i-...
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0answers
55 views

When does taking the product $- \times X$ preserve reflexive coequalizers in $\mathsf{Top}$?

By $\mathsf{Top}$ I mean the category of topological spaces and continuous homomorphisms. I am aware that if $(X,\tau_X)$ is a topological space, then the functor $- \times (X,\tau_X) : \mathsf{Top} ...
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1answer
40 views

Verifying separation axioms for the product topology $\{0,1\}^J$

Is my proof efficient? I think I was able to verify the separation axioms but I am still not entirely sure. Any help is greatly appreciated! Thanks! $\def\R{{\mathbb R}}$ I wish to prove the ...
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0answers
13 views

Showing a the Set of Joint Probability Measures is Closed. Euclidean Space.

Please find a similar question here: Is the set of all joint probability measure closed? $\underline{Notation :}$ Let $\mathcal{P}_2(\mathbb{R}^n)$ be the space of borel probability measures on $\...
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3answers
28 views

Homeomorphism between a subspace of a product topology and one of the factors of product space

This is my first question on SE. I will try to be as clear as possible. I have this question as my homework in my topology course. We have to prove that $X_1 \times \{x_2\} \sim X_1$, where $X_1 \...
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0answers
36 views

Weak topology of $X\times [0,1]$ with respect to a family of subspaces

This problem appeared to me at first when studying topology of CW-complexes. I also needed ths fact when trying to prove that some graphs are contractible, when seen as complexes. Basically, I need to ...
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3answers
44 views

Neighborhoods in product spaces

Consider the infinite product space where each component is the real numbers with the Euclidean topology and $p = (1,1,1,...)$. Show or disprove that $\forall U \ni p, \exists t > 0, \forall s \in (...
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1answer
24 views

Homeomorphic to a product space

Consider $\mathbb{R^2}$ with usual topology. We define $\mathcal{S}$ as the equivalence relationship on $\mathbb{R^2}$ that identifies to one point all elements in $\mathbb{Q^2}$. I need to check if $...
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3answers
85 views

Alternative definition of product topology

In James Munkres's book, product topology on any many sets is defined based on projection mapping which is based on J tuple. This definition seems to be not very intuitive. Why isn't it defined as: ...
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2answers
34 views

Proving $R_L \times R_L$ is completely regular. Meaning $R_L \times R_L$ is an example of a space which is completely regular, but not normal

Can I please receive help/feedback on my proof for the problem below? Thank you $\def\R{{\mathbb R}}$ Prove that $\R_L \times \R_L$ is completely regular. This means $\R_L \times \R_L$ is an example ...
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1answer
49 views

Closure of a solid triangle in $\mathbb{R}_{(-)} \times \mathbb{R}_{\textit{finite complement}}$

Consider the product topology $\mathbb{R}_{(-)} \times \mathbb{R}_{\textit{finite complement}}$. What is the closure of a solid triangle in this space? I know that if we consider the topological ...
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1answer
25 views

Proving the closed graph theorem in general topology

Let $X$ and $Y$ be topological spaces where $Y$ is compact and Hausdorff. Then let $f:X\rightarrow Y$ be a function and $G_{f}=\{(x,y)\in X\times Y|y=f(x)\}$ be the graph. The theorem is stated as ...
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1answer
17 views

Reference request on functions defined on Cartesian products

The first time I come across the functions defined on Cartesian product is when I read this: As indicated in the above picture, the facts 2 and 3 are straightforward and I can prove it. However, ...
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2answers
74 views

Projection of a closed set in $\mathbb{R}^2$ to one of the factors need not be closed.

$\def\R{{\mathbb R}}$ May I please receive help with the following problem? It is from Munkrees Topology Textbook. We know the projection to $X$ or $Y$ of an open subset of $X\times Y$ with the ...
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3answers
32 views

Let $X$ and $Y$ are topological spaces with indiscrete topologies then prove that the product topology $X\times Y$ will be indiscrete space

Now $\tau_X=\{X,\emptyset\}$ and $\tau_Y=\{Y,\emptyset\}$ their product topology will be like $\tau_{X\times Y}=\{X \times Y , \emptyset \times Y , X \times \emptyset , \emptyset\}$ which is clearly ...
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1answer
15 views

product-topology on $\mathbb{R} \times X$

Let $(X,||\cdot||_X)$ be a Banach space and $|\cdot|$ the absolute value on $\mathbb{R}$. How to show that $||(t,x)||:=|t|+||x||_X$, $(t,x) \in \mathbb{R} \times X$ induces the product Topology?
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1answer
24 views

Equivalence of Product Metric and $d_1$ Metric

I'm trying to show that the product metric $d_{X \times Y}((x_1,y_1),(x_2,y_2))=\max{\{d_X(x_1,x_2),d_Y(y_1,y_2)\}}$ and the metric $d_1((x_1,y_1),(x_2,y_2))=d_X(x_1,x_2)+d_Y(y_1,Y_2)$ define ...
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1answer
53 views

Separability of $\mathbb R^{[0,1]}$ in the product topology

Munkres has the following exercise: Show that the product space $R^I$, where $I=[0,1]$, has a countable dense subset. If $J$ has cardinality greater than $2^\mathbb{N}$, then the product space $\...
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0answers
45 views

Uniform Convergence on Standard Probability Space

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a standard (or Lebesgue-Rokhlin) probability space. That is, $(\Omega,\mathcal{F},\mathbb{P})$ is isomorphic to $[0,1]^{\mathbb{N}_{0}}$ endowed with the Borel ...
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1answer
128 views

Characterizing measurability in product spaces

Let $X,Y$ be non-empty sets. Define $$\Phi:\mathbb{R}^{X\times Y}\to ({\mathbb{R}^{Y}})^X, f\mapsto\left(x\mapsto\left(y\mapsto f(x,y)\right)\right).$$ Let $\mathcal{F}_X$ be a $\sigma$-algebra of ...
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1answer
50 views

Topological name for $D^m \times \mathbb{R}^n$

Is there a name for a topological space homeomorphic to $D^m \times \mathbb{R}^n$, where $D^m$ is a closed $m$-dimensional ball? I would call it $(m,n)$-cylinder if there is no other conventional ...
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1answer
53 views

Open sets in the infinite product of $\mathbb{N}^\infty$

How can I see that the open sets in the infinite product of $\mathbb{N}^\infty$ in the product topology with all components being the discrete topology are exactly the sets extending the given finite ...
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4answers
72 views

Topological space which is homeomorphic to its square

If $X$ is an infinite set, we can consider it as a topological space with the discrete topology, and it has the property that $X$ and $X\times X$ are homeomorphic. Does the property that $X$ is ...
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1answer
27 views

Example Problem from “Lecture Notes on Elementary Topology and Differential Geometry” (Singer/Thorpe)

The following is an example problem from the above-mentioned text: "Let $J^+$ = [n, n is a positive integer] and let $I_n = [0, 1/n]$. Then $P = (\pi_{n \; \in \; J^+}, I_n)$ is a topological space ...
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1answer
43 views

Homeomorphism to graph

Suppose $f:X\rightarrow Y$ is a continuous map. Define the $\textbf{graph of f}$ to be $\Gamma(f)=$ $\{$ $(x,y)$ $\in X\times Y$ $:$ $y=f(x)$ $\}$ . Then the map $\phi_f: X \rightarrow \Gamma(f)$, ...
8
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3answers
184 views

Are $(0,1) \times [0,1)$ and $[0,1) \times [0,1)$ homeomorphic?

I do not know how to approach. I tried removing points and see if the remaining spaces are connected or not, but i couldn't conclude anything without doubt.
10
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1answer
289 views

A sequence with no converging subsequence that clusters everywhere

Let $I = [0,1]$ be the compact unit interval and $T = I^I$ the Tychonoff cube. It is pretty standard to exhibit a sequence in $T$ with no convergent subsequence. It is also fairly standard to show ...
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2answers
61 views

$A$ is homeomorphic to $A\times A$ [duplicate]

Is there any infinite topological space $A$ which is connected such that $A$ and $A\times A$ are homeomorphic?
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1answer
36 views

Visualizing: “Altering a finite set of coordinates cannot change the value of a continuous function $f:X \to \{0,1\}$”

Okay, so this is lemma 2.11 in chapter 4 of Mendelson's Introduction to Topology (p. 117): Let $\{{X_\alpha\}}_{\alpha \in I}$ be an indexed family of topological spaces, each of which is connected....
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0answers
24 views

$\Bbb R^J$ is a Baire space in product topology

I have written the following proof of the fact that, $\Bbb R^J$ is Baire space in product topology. Can anyone check my proof and say about any fault? Thanks in advance. $\textbf{Proof :---}$ ...
2
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1answer
80 views

Why is $2^{\mathbb{N}}$ Homeomorphic to $2^{\mathbb{N}^{< \mathbb{N}}}$?

This question is based off a problem from Classical Descriptive Set Theory by Kechris. In this book, Kechris makes the claim that, when $\{0, 1\}$ is endowed with the discrete topology, then the ...
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2answers
92 views

Can $\mathbb{R}^2$ be homeomorphic to $\mathbb{R}\times\mathbb{R}$ with this topology?

Consider $\mathbb{R}^2$ with the included point topology (open subsets are those containing $(0,0)$); and $\mathbb{R}$ with the same topology (open subsets are those containing $0$). Now consider the ...
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0answers
48 views

A converse version of Fubini's Theorem

Assume two $\sigma-$finite measure spaces $(X,\mathcal{M},\mu)$ and $(Y,\mathcal{N},\nu)$. Consider a $\mathcal{M}\times\mathcal{N}$-measurable function $f$, and we are interested in computing $$ \...
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1answer
44 views

Show that $ \ g_2: \Bbb R \to \Bbb R$ given by $g_2: x \to x^{-1}$ (inverse map) is not continuous.

Let $(\Bbb R,+)$ be given the semi-open topology, i.e., the topological basis consists of open sets like $\{ [a,b)\}$. Then show that, $(i) \ g_1: \Bbb R \times \Bbb R \to \Bbb R$ given by $(x,y) \...
2
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3answers
142 views

Doubt about open sets in the Product topology

So we know that in the product topology a basis element is gonna be of the form $\prod U_\alpha$ where $ U_\alpha \neq X_\alpha $ for a finite number of $\alpha$. So my thing is that we know that a ...
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2answers
23 views

Box topology and a non-continuous function

I'm having trouble understanding box topologies on infinite spaces. I'm asked to prove that the function f going from ([0,1], standard top.) to ([0,1]^N, box top.) is not continuous. This function ...
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0answers
43 views

Absolute continuity with respect to the product measure

Let $\mu$ be a measure over the product space $X\times Y$, $X$ is any topological space and $Y$ is either Lindelöf or compact, and let $\mu_y$ and $\mu_x$ denote the marginals of $\mu$ on $X$ and $Y$, ...
3
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1answer
63 views

Expectation as product measure

I am reading a result that for a nonnegative random variable $X$ on $(\Omega, \mathcal{F}, P)$, $EX = (P \times \lambda)\{(\omega,x): 0 \leq x \leq X(\omega)\}$, where $\lambda$ is the Lebesgue ...
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0answers
46 views

How to construct metric of direct sum of spaces?

Consider the space $\mathbb{R}^3\oplus \mathrm{SU}(2)$. How do you construct a metric for it using the metrics for the subspaces? I'm considering using something like: $d(x_1, x_2) = d_1\left(\vec{v}...
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1answer
31 views

Trouble understanding open sets in product topology.

I have the following doubt. Consider a manifold $(M,\tau)$ and its product topology $\tau^2$. How is then an open set $U\in\tau^2$ defined? Is it $U=\bigcup_{i\in I} U_i\times V_i$, where $U_i,V_i\...
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1answer
53 views

If a product space is metrisable, are the factor spaces metrisable?

Let $((E_i,\mathcal O_i))_{i\in I}$ be an at most countable family of topological spaces, each containing at least two distinct elements. Let $E=\prod_{i\in I}E_i$ and $\mathcal O$ be the product ...
2
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1answer
52 views

Stuck on part of a proof: finding neighbourhood of diagonal with second coordinate satisfying a certain property

My friend was reading a proof from "An Introduction to Dynamical Systems" from Michael Brin, and he got stuck on a part of the proof that boiled down to the following: Let $X$ be a compact ...
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1answer
59 views

Why is $\{X=Y\}$ measurable if the diagonal is measurable?

Suppose $X$ and $Y$ are random variables on $(\Omega, \mathcal{F})$ with values in $(\mathcal{X},\mathcal{A})$. I would like to prove that if the diagonal $$\Delta:=\{(x,y)\in \mathcal{X}^2:x=y\}$$ ...
2
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1answer
62 views

Is equivalence of probability measures preserved under infinite products?

For all $n\in \Bbb N$, let $\mu_n$ and $\nu_n$ be equivalent probability measures on a measurable space $(\Omega_n,\mathcal{F}_n)$. Are $$ \mu:=\bigotimes_{n=1}^\infty\mu_n \quad \text{and} \quad \nu:...
2
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1answer
41 views

How come an element $s\in (\varprojlim S_i)^c$ does not obey $\pi_{ji}(s_j)=s_i$?

Notation: 1. $S_i$ is a hausdorff space for every $i\in \mathbb N$ 2. $(S_i, \pi_{ji})$ is an inverse system where $\pi_{ji}:S_j\rightarrow S_i$ 3. $S=\varprojlim S_i $ I wish to show $S$ is a ...

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