For questions about the structure of product space, in the context of topology or measure theory for example. Use other tags to indicate the context.

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-2
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1answer
46 views

How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
0
votes
1answer
54 views

Probability of Union of Events in a Probability Product Space By Counting Event Size

This question is about probability of union of events in a probability product space. Let’s say a fair die is thrown twice and we’re interested to find out probability of getting face value one in 1st ...
2
votes
3answers
49 views

If $(X,\tau)$ is Hausdorff, is $X\times X$?

If $(X,\tau)$ is Hausdorff, is the product topology $X\times X$? I have a feeling that some part of $\Bbb R$ to be Hausdorff, means there is isolation for all elements on the line, and if that is the ...
5
votes
1answer
48 views

Compact subsets of the space of real functions $\mathbb{R}^\mathbb{R}$

I was suprised that this question hasn't been asked - or maybe it was, but asked differently. Anyway, I want to characterize the compact sets in the space of real functions $\mathbb{R}^\mathbb{R}$ ...
1
vote
1answer
30 views

Continuity of product maps

Let $J$ be a given (countably or uncountably infinite) index set. Let $\{\ X_\alpha \ \colon \ \alpha \in J \ \}$ and $\{\ Y_\alpha \ \colon \ \alpha \in J \ \}$ be collections of topological ...
0
votes
1answer
54 views

On the Preservation of Product Measurability under Partial Conditional Expectation.

Let $(X,\mathcal{X},\mu)$ and $(Y,\mathcal{Y},\nu)$ be probability spaces, $\mathcal{X}_{0}\subset\mathcal{X}$ a (sub)sigma field and assume that $f=f(x,y)\in L^{1}_{\mu\otimes \nu}$ where $(X\times ...
1
vote
0answers
40 views

$(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$

Let $X_1$, $X_2$, and $X_3$ be spaces. (a) Prove that $(X_1 \times X_2) \times X_3$ is homeomorphic to $(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$ So, I think I ...
1
vote
0answers
22 views

Continuity of Product Topology [duplicate]

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a ...
1
vote
0answers
52 views

Topological Equivalence of Product Metric Spaces

Suppose that the metric space $(X_i,d_i)$ is topologically equivalent to $(Y_i,d'_i)$ for $i=1,2, \cdots , n$. Show that the product metric spaces $X = \prod_{i=1}^nX_i$ and $Y= \prod_{i=1}^nY_i$ are ...
0
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1answer
37 views

Every section of a measurable set is measurable? (in the product sigma-algebra)

I'm studying measure theory, and I came across a theorem that says that, given two $\sigma$-finite measure spaces $\left(X,\mathcal M,\mu\right)$ and $\left(Y,\mathcal N,\nu\right)$, and a set $E\in ...
2
votes
1answer
51 views

Questions about shift-invariant measures in ${\bf N}$

Let $P$ be a shift invariant diffused probability measure defined on powerset of all natural numbers ${\bf N}$(see, van Douwen, Eric K. (1992). Finitely additive measures on ${\bf N}$. Topology ...
2
votes
0answers
40 views

A question about product $\sigma$-algebra

Assume $(X,\mathscr{G})$ is a measurable space where $X$ is a finite set and $\mathscr{G}$ is just the power set. Let $\Omega$ be the infinite product of $X$ and $\mathscr{F}$ be the usual product ...
2
votes
0answers
37 views

Show that iterated integral of indicator function over product space defines a countably additive (product) measure.

Let $\Sigma = \Sigma _2 \times \Sigma _2 $ Given that $x $-and $y $ sections are $\Sigma _1 $ - and $\Sigma _2 $ measurable respectively, and that for $F \in \Sigma $, $\int ( \int I _F \, d \mu _1 ...
3
votes
1answer
80 views

Product measures and $\sigma-$ finite measures

Problem similar to folland chapter 2 problem 51. The actual problem in Folland mentions that $X,Y$ are not necessarily $\sigma-$finite. Then how can I use Fubini-Tonelli theorem?
2
votes
0answers
35 views

Product measures in measure theory

Suppose $(X,M,\mu)$ and $(Y,N,v)$ are $\sigma-$finite measure spaces. Is the following calculation correct? $\int \chi_E(x,y) d\mu=\int (\chi_E)^y(x) d\mu=\int \chi_{E^y}(x) d\mu=\mu(E^y)$ for a ...
3
votes
2answers
73 views

Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the product topology on $\mathbb{R}^m \times \mathbb{R}^n$

I'm attempting to teach myself topology for graduate school this summer, but I'm having a tough time. I'm trying to prove that the Euclidean topology on $\mathbb{R}^{m+n}$ is equivalent to the ...
2
votes
2answers
285 views

Infinite product of probability measures is a premeasure

This is an exercise from Real Analysis by Stein and Shakarchi (Chapter 6, Exercise 15). Given infinitely many measure spaces $(X_i, \mathcal M_i, m_i)$, each of which has measure 1, one can define an ...
2
votes
2answers
226 views

The topology a line inherits as a subspace of $\mathbb{R}_l \times \mathbb{R}$, or of $\mathbb{R}_l \times \mathbb{R}_l$ (Munkres)

Exercise 8, Chapter 2, Pag. 92, from the Munkres's book: The topology $\mathbb{R}_l$ is the topology on $\mathbb{R}$ generated by all half-open intervals of the form \begin{align*} [a, b) = \{ x \, ...
4
votes
3answers
98 views

If $(X,\tau)^n$ is Hausdorff, is $(X,\tau)$ also Hausdorff?

If $(X,\tau)^n$ is Hausdorff, is $(X,\tau)$ also Hausdorff? I know that product of Hausdorff space is Hausdorff, but I want to know if this weaker converse of it is true. Thanks.
1
vote
0answers
69 views

Lebesgue-integrable and existence of integral

I have given the following function $$ f(x,y) = \begin{cases} 1 &, \ x \in \mathbb{Q} \\ 2y & , \text{ otherwise} \end{cases} $$ This is a measurable function in sense of Lebesgue. Now, I ...
4
votes
1answer
162 views

Questions on Fubini's Theorem and $\sigma$-finite measure?

I asked a question about this a several days ago, but I think I have a better formulated question now. The reason I did not just edit the last question about this is that I feel the answers I got ...
1
vote
1answer
31 views

The general form of a measurable set in a product measurable space

Hi everyone: Suppose that $(X,\mathfrak{M},\mu)$ and $(Y,\mathfrak{N},\nu)$ are two measure spaces and consider the product measure space $(X\times ...
4
votes
1answer
152 views

Product of T1 spaces is T1

I am trying to prove that the product of T1 spaces is also T1. Here is a proof, is it correct? $\{ X_i \}_{i \in I}$ are T1 $\Rightarrow$ $\prod_{i \in I} X_i$ is T1 Proof: Let $\bar{x} = ( ...
1
vote
1answer
42 views

How to find subbase and base for $X\times Y$?

Let $\tau :=\{X,\emptyset,\{a\},\{b,c\}\} $ on $X=\{a,b,c\}$ and $\tau^*:=\{Y,\emptyset,\{u\}\}$ on $Y:=\{u,v\}$ i) Find a subbase for the product topology on $X\times Y$ ii) Find a ...
0
votes
1answer
20 views

continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous

Say $X$, $Y_1$ and $Y_2$ are topological spaces. Let $f_1 \; X \to Y_1$ and $f_2 \; X \to Y_2$. If $f\; X \to Y_1 \times Y_2 $ $f(x) = (f_1(x), f_2(x))$ $Y_1 \times Y_2$ is a topological space with ...
0
votes
1answer
43 views

Prove $ \varphi ( x ) = \lambda ( Q_x ) $ is $ \mathcal{X} $-measurable

I'm reading Thoerem 8.6, page 163, Chapter 8 from "Real and Complex Analysis (Third Edition)" by Walter Rudin; it seems that Rudin never proves that $\lambda(Q_x)$ (defined below) is a ...
2
votes
2answers
254 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
2
votes
1answer
192 views

A generalization of the generalized tube lemma

I am trying to prove the following generalization of the generalized tube lemma: Let $\{X_t\}_{t \in T}$ be a family of Hausdorff spaces and $\prod_{t \in T}A_t$ be a compact subset of $X=\prod_{t ...
1
vote
1answer
153 views

What is the definition of a product topology?

I am new to advanced mathematics and I recently started reading a book on topology. I am struggling to understand what it is saying in this paragraph. This is what it says: Let $E_i$ ...
4
votes
2answers
241 views

Product topology on $X \times Y$ the smallest topology when $f(x, y) = x$ and $g(x, y) = y$ are continuous functions?

$X$ and $Y$ are topological spaces and let $f : X \times Y \to X$ and $g : X \times Y \to Y$ be maps such that $f(x, y) = x$ and $g(x, y) = y \ \forall (x, y) \in X \times Y$ . Show that the product ...
1
vote
1answer
71 views

Prove that the projection $p_i$ from the product space $X_I S_i \tau$ into $S_i \tau_i$ is open for each $i \in I$

Full question: A function $f$ from a space $X, \tau$ to a space $Y, \tau'$ is said to be open if $f(V)$ is open in $Y$ whenever $V$ is an open subset of $X$. Prove that the projection $p_i$ from the ...
2
votes
2answers
44 views

Show that is not a sigma algebra

I would like to pose a question to you, and I will appreciate any hint or partial solution for my question. Let $\left(\Omega, \mathcal{F}\right)$ be a measurable space, and let $\left(\Omega \times ...
2
votes
0answers
65 views

Is $\sigma$-finiteness necessary for the existence and uniqueness of product measure?

Let $(X,\mathfrak{B}_X,\mu_X)$ and $(Y,\mathfrak{B}_Y,\mu_Y)$ be $\sigma$-finite measure spaces. Then there exists a unique measure $\mu_X\times\mu_Y$ on $\mathfrak{B}_X\times\mathfrak{B}_Y$ that ...
3
votes
0answers
36 views

Right-continuity of filtrations on product spaces

Let $(\Omega^1, \mathcal{F}^1)$ and $(\Omega^2,\mathcal{F}^2)$ be two measurable space and let $(\mathcal{F}^2_s)_{s \geq 0}$ be a filtration on $(\Omega^2,\mathcal{F}^2)$. Moreover, let $t\geq 0$ be ...
3
votes
2answers
614 views

Convergence in product topology

Let $x_1,x_2, \ldots$ be a sequence of points of the product space $\prod X_\alpha$. Show that the sequence converges to the point $x$ if and only if the sequence $\pi(x_1), \pi(x_2)\ldots$ converges ...
4
votes
3answers
186 views

Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$.

Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$. When I sketch these spaces it this statement makes sense intuitively because $[0, 1]\times[0, ...
3
votes
1answer
1k views

The product of Hausdorff spaces is Hausdorff

I'm confused how it can be true that the product of an infinite number of Hausdorff spaces $X_\alpha$ can be Hausdorff. If $\prod_{\alpha \in J} X_\alpha$ is a product space with product topology, ...
2
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1answer
484 views

A product of two sequentially compact metric spaces is compact. How to prove this explicitly?

We know that a product of two (or finitely many) compact topological spaces is compact. And we also know that in a metric space, compactness is equivalent to sequential compactness. So a product of ...
3
votes
2answers
258 views

Is projection of a closed set $F\subseteq X\times Y$ always closed?

If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not ...
2
votes
2answers
145 views

Is this a homeomorphism?

Suppose you have a cartesian product of spaces $\prod_{\alpha\in\mathcal{A}}X_{\alpha}$ in the product topology. Choose any $\alpha\in\mathcal{A}$ . Is the following a homeomorphism of a subspace ...
3
votes
2answers
302 views

Projection of a closed subspace [duplicate]

If we have two topologies $(X,\mathcal{T})$ and $(Y,\mathcal{U})$, then we may take the product topology. We define the projection map $\prod_x$ in the usual way. If $A\subset X\times Y$ is closed, ...
5
votes
1answer
81 views

Is $(A \times B)^\epsilon \subseteq A^\epsilon \times B^\epsilon$?

While working on a problem related to my research, I had the following query. It pertains to product spaces: The Question: Let $(X,d_X)$ and $(Y,d_Y)$ be two Polish (Complete separable metric) ...
1
vote
1answer
534 views

Projection mapping closed in compact space [duplicate]

Consider a topological space $(X, \mathcal{T})$. Suppose $X$ is compact and $(Y, \mathcal{T}_Y)$ is Hausdorff. Let $\Phi: X \times Y \rightarrow Y$ be the projection map. We show that $\Phi$ is a ...
1
vote
1answer
87 views

Are $\prod_{i\in I,j\in J}X_{ij}$ and $\prod_{i\in I}\prod_{j\in J}X_{ij}$ homeomorphic?

For each $i\in I, j\in J$, $X_{ij}$ is a topological space. Are $$\prod_{i\in I,j\in J}X_{ij}$$ and $$\prod_{i\in I}\prod_{j\in J}X_{ij}$$ homeomorphic? What's the homeomorphism?
6
votes
1answer
437 views

Does factor-wise continuity imply continuity?

Let $f$ denote a map from a product space $X \times Y$ to $Z$. If for every $x\in X$, the map $f(x,-)$ is continuous, and the same holds for every $y \in Y$, then is $f$ continuous in general? If not, ...
2
votes
1answer
309 views

If product $X_1\times X_2$ and $Y_1\times Y_2$ are homeomorphic is $X_i\simeq Y_i$?

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
1
vote
1answer
246 views

Definition of product of uniform spaces

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous. But Springer's encyclopedia ...
7
votes
1answer
248 views

Continuous maps from products of topological spaces

Let $X,Y,Z$ be topological spaces. It is well-known that if $F:X\times Y\to Z$ is a continuous map, we can define a map $$\overline{F}:X\to C(Y,Z) \\\overline{F}(x)(y)=F(x,y)$$ where $C(Y,Z)$ is the ...
8
votes
1answer
256 views

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesn't possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ ...
11
votes
4answers
4k views

$X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed

Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$ Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$. ...