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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1answer
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Urysohn Metrization Theorem contradiction (uniform topology homeomorphic to product topology)?

The theorem states that if $F$ is regular and has a countable basis, then it is metrizable. In Munkres' proof of this theorem, he gives a function (homeomorphism) $F:X \rightarrow [0,1]^\omega$ that ...
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32 views

Lindelöf space with compact space [closed]

This is one of the exercises from Munkres'. Show that if $X$ is Lindelöf and $Y$ is compact, then $X\times Y$ is Lindelöf.
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1answer
45 views

Fubini's theorem for a measure on product space, which is not a product of measures

Let $X,Y$ be some nice measurable spaces (I'm interested in $[0,1]$ so we can assume compact, etc.). Let $\mu$ be a measure on $X\times Y$ (again, assume it's a nice probability measure, or even ...
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3answers
246 views

Question about continuity in the box topology

I have two question regarding the following example in Munkres (1)Why "if $f^{-1}(B)$ were open it would contain some interval $(-\delta,\delta)$ about 0. (2)My second question is somewhat broad, but ...
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2answers
40 views

Box topology definition

Munkres defines the box topology as shown below. I am trying to understand how come an element of basis as defined is even subset of $\prod_{\alpha \in J}X_{\alpha}$. If we get an element of the basis,...
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0answers
23 views

Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...
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1answer
20 views

Showing a space is homeomorphic to the Hilbert cube.

Let $(X_i,T_i)$ be a countably infinite family of topological spaces each of which is homeomorphic to the Hilbert cube. Show that $\prod_{i=1}^{\infty}(X_i,T_i) \cong I^{\infty}$. The question also ...
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1answer
58 views

Question about the proof that the Hilbert Cube is compact.

Because of the fact that $(1)$ The topological space $[0,1]$ is a continuous image of the Cantor space $(G,T)$. There exists a mapping $\phi_n$ of $(G_n, T_n)$ onto $(I_n, T'_n)$ where, for each ...
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1answer
51 views

Proof that a mapping onto $[0,1]$ is continuous.

Let each $(A_i,T_i) = (\{0,2\}, T_{discrete})$ and define $\phi : \prod (A_i, T_i) \rightarrow [0,1]$ with $\phi (<a_1, a_2, ...>) = \sum^{\infty}_{i=1} \frac{a_i}{2^{i+1}}$. In order to show ...
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1answer
29 views

Mapping of open subsets of product spaces.

Let each $(A_i,T_i) = (\{0,2\}, T_{discrete})$ and define $\phi : \prod (A_i, T_i) \rightarrow [0,1]$ with $\phi (<a_1, a_2, ...>) = \sum^{\infty}_{i=1} \frac{a_i}{2^{i+1}}$. If we let $W = \{...
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0answers
32 views

Question about: Prove that $\phi : \prod_{i=1}^{\infty} (A_i,T_i)$ onto $[0,1]$ is continuous.

My question is: How does choosing $N$ sufficiently large effect this proof? Prove that $\phi : \prod_{i=1}^{\infty} (A_i,T_i)$ onto $[0,1]$ is continuous. Let each $(A_i,T_i) = (\{0,2\}, T_{...
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1answer
37 views

The topological space $[0,1]$ is a continuous image of the Cantor space question.

Prove that the topological space $[0,1]$ is a continuous image of the Cantor space $(G,T')$. I know that this means to show there exists a function $$(i) f : (G,T') \rightarrow [0,1]$$ such that $f$ ...
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1answer
25 views

If a product space is locally compact, then each space is locally compact and all but a finite number of factors are compact

If $\prod^{\infty}_{i=1} (X_i, T_i)$ is locally compact, then each $(X_i, T_i)$ is locally compact and all but a finite number of $(X_i, T_i)$ are compact. Let $X=\prod^{\infty}_{i=1} (X_i, T_i)$, ...
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0answers
34 views

Open sets in a discrete product space.

How is it possible for a countably infinite product space to be discrete? If we let each $(X_i,T_i)$ be topological spaces with more than $1$ point and $$\prod^{\infty}_{i=1}(X_i,T_i)$$ be the ...
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3answers
43 views

How can a countably infinite product space be discrete?

Let $(X_i,T_i), i \in \Bbb N$, be a countably infinite family of topological spaces. Prove that $\prod^{\infty}_{i=1} (X_i,T_i)$ is a discrete space iff each $(X_i, T_1)$ is discrete and all but a ...
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0answers
25 views

If each $(X_i,T_i)$ is finite and discrete, then the box product is not compact.

If each $(X_i,T_i)$ is finite and discrete, then the box product is not compact. Let each $(X_i,T_i) = \{x_i\}$, then the box product is only covered by the open set $U = \prod \{x_i\}$, but since ...
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1answer
32 views

Product spaces from metric induced topological spaces $(X,T) \times (X,T)$ into $\Bbb R$ is continuous

Let $(X,d)$ be a metric space and $T$ the induced topology on $X$ by $d$. Prove that the function $d$ from the product space $(X,T) \times (X,T)$ into $\Bbb R$ is continuous. My question is what ...
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1answer
64 views

Show that $X \times Y$ is homeomorphic to $Y \times X$

This seems to be a trivial result, I just wish to check for correctness. Show $X \times Y$ is homeomorphic to $Y \times X$ Let $\tau$ be the topology on $X$, $j$ be the topology on $Y$ Then a ...
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1answer
48 views

Norms on an Ultraproduct

Suppose $X$ is a Banach space and $\mathcal{U}$ is a non-principal ultrafilter on $\mathbb{N}$. I am interested in the Banach space $(X)_\mathcal{U}$, where we consider sequences $(x_i)_{i \in \mathbb{...
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0answers
28 views

Example Infinite Product of Compact Metric Spaces that is not itself compact [closed]

Can anyone think of an infinite product space of compact metric spaces that us not itself compact?
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30 views

Second derivative/tangential map on a product manifold

Short version: Given $f: M_1 \times M_2 \longrightarrow N$ what is the second tangential $TTf$ expressed in partial tangentials on $M_1, M_2$? The details: Let $M_1, M_2, N$ be manifolds. The partial ...
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2answers
49 views

Showing that a mapping of a topological space into a product space is continuous.

Let $(Y,T)$ and $(X_i,T_i)$, $i = 1,...,n$ be topological spaces. Further for each $i$, let $f_i$ be a mapping of $(Y,T)$ into $(X_i, T_i)$. Prove that the mapping $f: (Y,T) \rightarrow \prod (X_i,...
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2answers
36 views

If $\prod (X_i,T_i)$ is connected, then each $(X_i, T_i)$ is connected.

Let $(X_i,T_i)$ be a topological space and $i=1, 2, ..., n.$ If $\prod (X_i,T_i)$ is connected, then each $(X_i, T_i)$ is connected. I attempt a proof by contrapositive. Let $U_i$ and $V_i$ be two ...
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1answer
71 views

What is a finite subcover of $[0,1]^{[0,1]}$?

According to Tychonoff's theorem, under the standard topology, $[0,1]^{[0,1]}$ is compact. However, I cannot think of a finite subcover of this space. Also, how does this reconcile with the fact that ...
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2answers
47 views

To show that product $Z=X×Y$ in the product topology is a CW complex

I would like to prove the following: If $X,Y$ are CW complexes, and either $X$ or $Y$ is locally compact then the product $Z=X×Y$ in the product topology is a CW complex. (-see here) In order to ...
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2answers
50 views

Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" $...
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1answer
38 views

How do I prove that the product topology is the unique topology for which projections are well-behaved?

This is a question from a practice qualifying exam. Let $X, Y, T$ be topological spaces. Define $p_X: X\times Y\to X$ and $p_Y:X\times Y\to Y$ to be the projection maps. Then $f:T\to X\times Y$ is ...
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4answers
397 views

In general, why is the product topology not equal to the box topology

I am trying to understand a counter-example showing that the box topology and product topology are not equal. Here it is: Let $\tau$ and $\tau'$ be the product and box topologies respectively. Let $...
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1answer
110 views

Borel measurability of a subset of a product space

Let $X$ and $Y$ be compact metric spaces and let $\mathcal B_X$ and $\mathcal B_Y$ be their respective Borel $\sigma$-algebras. Let $\mu$ be a Borel probability measure on $X$ and let $\mathcal B^*...
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0answers
9 views

General set in product space approximated by rectangle sets

Let $(E^k,\mathcal{E}^k,\mu^k)$ be a product measure space. By a rectangle set in $E^k$, we mean a set of the form $A_1\times\ldots\times A_k$ where each $A_i\in \mathcal{E}$. My question is, for ...
5
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1answer
93 views

Proving $(A\times B)^- = A^-\times B^-$ (closure of cartesian product)

My proof, for: $$(A\times B)^- = A^-\times B^-$$ using the metric $$d''((a_1,a_2),(b_1,b_2)) = max\{d_1(a_1,b_1),d_2(a_2,b_2)\}$$ $\rightarrow$ Well, if $a = (a_1,a_2)\in (A\times B)^-$ then: $$...
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1answer
59 views

Topological proof of the compactness of product metric space

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two compact metric spaces (see the definition here). Then show that the product metric space $(X\times Y,d_{X\times Y})$ is also compact. Now this can be done ...
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1answer
53 views

Show that $\operatorname{Cl}(A) \times \operatorname{Cl}(B) = \operatorname{Cl}(A \times B)$ without using idea of closure points

I've already shown that $\operatorname{Cl}(A \times B)$ is a subset of $\operatorname{Cl}(A) \times \operatorname{Cl}(B)$, but I'm not sure how to show the other containment: $\operatorname{Cl}(A) \...
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0answers
24 views

How is the absolute value on a product space defined?

I want to show that a map from the cantor set $C$ to its product $C\times C$ is homeomorphic, and I am stuck at continuity because in the epsilon delta proof I am not sure how to deal with the ...
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4answers
106 views

Show the Cantor set $C$ is equal to its product $C\times C$.

I've been reading up on the Cantor set, and it is simple to show a bijection $C \to [0,1]$. I was thinking that it would be easy to show that there exists some space filling curve by showing that $C =...
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0answers
22 views

Convergence in the product of spaces of iteratively composed functions.

My question is a bit odd, in fact conceptually it is not difficult, only that it operates on objects that are complex (to me). I would like to check two types of convergence in the product of the ...
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0answers
29 views

Product of Riemannian manifolds and volume element

Let $X$ and $Y$ be Riemannian manifolds and consider a function \begin{align} f\colon X\times Y &\to \mathbb{R},\\ (x,y) &\mapsto f(x,y) \end{align} Now I have to integrate $f$ with ...
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1answer
26 views

Is a subspace of a product space also a product space?

Suppose $V$ is a product space of connected and separable spaces $V_1,...,V_n$. Let $W$ be a subspace of $V$ with the subspace topology. Is $W$ then a product space of subsets of $V_1,...,V_n$? My ...
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48 views

Composition of two probability kernels is measurable

Let $p$ be a probability (or Markov) kernel with source $(X,\mathcal{A})$ and target $(Y,\mathcal{B})$, and $q$ a probability kernel with source $(Y,\mathcal{B})$ and target $(Z,\mathcal{C})$. We can ...
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3answers
85 views

Is a set contained in its Cartesian product with itself?

Given the Cartesian product $X = A \times A$, is $A \subset X$?
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1answer
41 views

Show that the area set is measurable

This is a problem from Billingsley's text Probability and measure. Suppose that $f$ is nonnegative on a $\sigma$-finite measure space $(\Omega, \mathscr{F}, \mu)$. Show that $$\int_\Omega f d \...
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1answer
47 views

Prove that d is a continuous real valued function on $M \times M$ [duplicate]

Let $(M,d)$ be a metric space. Let $M \times M$ be the product space, where $d$ is defined (earlier in the book as ) $d((x_1,x_2),(y_1,y_2))=d_1(x_1,y_1)+d_2(x_2,y_2)$ where $d_1$ and $d_2$ are ...
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1answer
23 views

Products containing the same digits

I am looking for $d_1$ and $d_2$, where $d_1$ is a two digit integer and $d_2$ is a three digit integer, such that $d_1\cdot d_2$ is a product that contains the exact same digits as $d_1$ and $d_2$. ...
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1answer
60 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 ...
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1answer
69 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 ...
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3answers
80 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
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1answer
53 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 $\mathbb{R}^\mathbb{R}$ (...
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1answer
39 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 spaces....
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1answer
62 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 Y,...
2
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0answers
54 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 ...