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.

Filter by
Sorted by
Tagged with
1
vote
0answers
27 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}...
0
votes
1answer
26 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\...
2
votes
1answer
51 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
votes
1answer
50 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 ...
1
vote
1answer
40 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
votes
1answer
50 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
votes
1answer
40 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 ...
1
vote
2answers
54 views

Inverse of a projection function

I have got the following question; Let $(X_1, \tau_1), (X_2, \tau_2)$ be topological spaces and $X = X_1 \times X_2$. Equip $X$ with the product topology $\tau$ so that, by definition of $\tau$, the ...
0
votes
1answer
29 views

Comparison between uniform and box topology on $\mathbb{R}^J$

Was reading Munkres' Topology and got stuck in this. He proves that uniform topology is contained in box topology on $\mathbb{R}^J$ but leaves the case that this inclusion is strict when $J$ is ...
1
vote
3answers
87 views

Countable product of metric spaces is metrizable [General Metric]

I know that if we have a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$ then $X=\Pi^{\infty}_{n=1}X_n$ is a metric space with metric $\rho((x_n)_{n \in \mathbb{N}},(y_n)_{n \...
2
votes
1answer
42 views

Is $\mathbb P((X,Y)\in A)=\int_{\mathbb R}\mathbb P(X\in A^y\mid Y=y)\mu_Y(dy)$ always true?

I know that if $\mu_1$ and $\mu_2$ are two measures on $\mathbb R$, then if $E\subset \mathbb R^2$, then $$\mu_1\otimes\mu_2(E)=\int_{\mathbb R}\mu_1(E^y)\mu_2(dy),$$ where $E^y=\{x\mid (x,y)\in E\}$. ...
0
votes
2answers
63 views

Prove spaces are homeomorphic, not that obvious.

So we are asked to prove that two sub-spaces of euclidean plane, namely $A=\mathbb{N}\times (\{\frac{1}{i}|i\in \mathbb{N}\}\cup\{0\})$ and $B=\mathbb{N} \times (\mathbb{N}\cup \{\frac{1}{i}|i\in \...
2
votes
0answers
37 views

What's the correct version of this “product rule” is topology? [duplicate]

An exercise asks me to do the following: $(x_1,\tau_1), (X_2,\tau_2)$ are two topological spaces. $A_i\in X_i, i=1,2.$ Show that $\partial(A_1\times A_2)=(\partial A_1\times A_2)\cup(A_1\times \...
0
votes
0answers
34 views

Topology of the automorphic quotient

The MO question https://mathoverflow.net/questions/331549/compactness-of-the-automorphic-quotient-and-genericity made me realise that I don't really understand the topology on the adèlic points of an ...
1
vote
1answer
42 views

Assume that Y is compact. Prove that f is continuous. [duplicate]

Let $f:X\rightarrow Y$ be any map. The graph of f is the set $\Gamma_f=\{(x,f(x))| x\in X\}\subset X \times Y$. Assume that Y is compact. Prove that if $\Gamma_f \subset X \times Y$ is closed then $...
0
votes
0answers
64 views

Product of two sequential spaces is not sequential

The product of two sequential topological spaces is not necessarily sequential. Could you give an example confirming this? As each first-countable space is sequential and the operation of taking a ...
0
votes
1answer
18 views

Product topology. Clarification needed

Please consider the statement and it's proof below: The author further says that : Members of the product topology can all be expressed as union of products, but most members of the product topology ...
1
vote
1answer
34 views

Show $g$ integrable over [0, 1] if $g(x) - g(y)$ is Lebesgue integrable over $[0, 1] \times [0, 1]$

Let $g$ be a Lebesgue measurable function on $[0, 1]$ such that the function $f(x, y) := g(x) - g(y)$ is Lebesgue integrable over the square $[0, 1] \times [0, 1]$. Show that $g$ is integrable over $[...
0
votes
0answers
3 views

Finite product of two Hurewicz space

X is called Hurewicz if X satisfies $U_{fin}(\mathcal{O},\Gamma)$. Example of 2 Hurewicx spaces whose product is not Hurewicz.
0
votes
1answer
20 views

Continuity of countable projection from non-first countable topological space

This might be trivial, but I just want to make sure I got this right: Let $X$ be a metric space and $I$ an uncountable index set. Let us consider $X^I$ with the product topology (of course, the ...
1
vote
1answer
29 views

Does a sequentially continuous function take its supremum on compacts?

Consider the following situation: Let $X$ be a separable metric space [if this helps: I am mainly interested in the case $X = \mathcal{P}(\mathbb{R}^d)$, the space of all Borel-probability measures on ...
4
votes
1answer
39 views

When is the product of closed sets closed in the product topology?

I have a specific example below and i think my proof is wrong because it seems too simple, it would work for the general case which I doubt is true. I would also be interested in the general answer ...
1
vote
2answers
76 views

$+:X\times X\to X,(x,y)\mapsto +(x,y)=x+y$ and $\cdot:\Bbb{R}\times X\to X,(\lambda,x)\mapsto \cdot(\lambda,y)=\lambda\cdot x$ are weakly continuous

$$+:X\times X\to X,\\(x,y)\mapsto +(x,y)=x+y$$ and $$\cdot:\Bbb{R}\times X\to X,\\(x,y)\mapsto \cdot(\lambda,y)=\lambda\cdot x$$ are weakly continuous, where $X$ is an infinite dimensional normed ...
0
votes
1answer
31 views

Cellularity of product space

Let $\{X_s\}_{s\in S}$ be a family of topological space, and $d(X_s)\leq m$, then the cellularity, i.e., the supremum of the cardinalities of all families of pairwise disjoint non-empty open subsets ...
0
votes
1answer
85 views

Proof Verification: Show that if Y is compact, then the projection is a closed map

I searched the site for other proofs that may be similar to mine and couldn't find one. I was hoping someone could review my variation in particular for correctness. Thanks in advance. Problem: ...
1
vote
0answers
52 views

Every bounded net in a dual space has a cluster point?

Question: Let $X$ be a normed space and $X^*$ be its continuous dual of $X.$ Assume that $(x_\alpha^*)_\alpha$ is a bounded net in $X^*.$ Is it true that there exists a cluster point $x^*$ in $X^*$ ...
1
vote
1answer
50 views

Is the subspace topology of the product topology the initial topology with respect to the restrictions of the projection maps?

Let $I$ be an index set, $X_i$ be a topological space for each $i \in I$ and $X = \prod_{i \in I} X_i$ the product of all $X_i$. Then the product topology is exactly the initial topology with respect ...
0
votes
1answer
20 views

How do I prove the $f:X\to I$ by $f(x)=\min\{f_i(x_{\beta_i}):i=1,2,…,n\}$ is continuous.

How do I prove that $f:X\to I$ by $f(x)=\min\{f_i(x_{\beta}):i=1,2,...,n\}$ is continuous. My attempt:- Minimum of two continuous real-valued functions are continuous. $$f_i \circ \pi_{\beta_i}: X\...
0
votes
1answer
32 views

Is $D$ closed in $\Bbb R^n$? [closed]

Let $C$ be a non-empty closed subset of $\Bbb R^n$ and $C × D$ be a closed subset of $C × \Bbb R^n$. Can we say that $D$ is closed in $\Bbb R^n$?
13
votes
0answers
197 views

Does the box topology have a universal property?

Given a set of topological spaces $\{X_\alpha\}$, there are two main topologies we can give to the Cartesian product $\Pi_\alpha X_\alpha$: the product topology and the box topology. The product ...
0
votes
1answer
61 views

Product of a compact topological space and a singleton in another topological space is compact proof

It's before we prove that 'Product of two compact sets is compact'. There are topological spaces $X$(which is compact), $Y$ and the product topology on $X \times Y$ is given by the subbase $U \times V$...
0
votes
2answers
63 views

Product of a compact set and a singleton is compact proof

It's before we prove that 'Product of two compact sets is compact'. Let $S$ be an open cover of $X \times \{\bullet\}$ where $X$ is compact. Then $\pi_1(S)$ is an open cover of $X$ so there is a ...
1
vote
1answer
20 views

Proving there exists $V$ so that $\{x_0\}\times I\subset V\times I\subset U$

Let $X$ be a topological space, and $I=[0,1]$. Consider $X \times I$ with the product topology. Now fix $x_0\in X$ and $U\subset X\times I$ an open that contains $\{x_0\}\times I$. Prove that there ...
3
votes
2answers
67 views

Question regarding proof of Tychonoff's theorem

On Wikipedia it states that a space $X$ is compact if and only if every net has a convergent subnet. It then states that a net in the product topology has a limit if and only if each projection has a ...
2
votes
0answers
37 views

Is $[0,1]^A$ not sequentially compact for any uncountable $A$?

We know $\{0,1\}^\mathbb{N}$ is sequentially compact, but $\{0,1\}^\mathbb{R}$ is not. The proof I have seen that the second is not sequentially compact heavily relies on the fact that we have a ...
1
vote
1answer
35 views

Questions about Tychonoff spaces natural embedding.

On Wikipedia it states that for any Tychonoff space $X$ there is a natural embedding into $[0,1]^{C(X,[0,1])}$. I assume this embedding is $\iota(x)(f)=f(x)$. I am able to prove that $\iota$ is ...
3
votes
0answers
51 views

Do separately semi-continuous functions have a dense set of semi-continuities?

The connection between separate continuity and joined continuity has been studied quite a lot. In particular, one has (as a special case of a far more general Theorem from here) the following: If $...
1
vote
1answer
54 views

Proving $(-1,1)^{\mathbb{N}}$ is not open in the product topology of $\mathbb{R}^{\mathbb{N}}$

Clarification: here $\mathbb{R}^{\mathbb{N}} = \mathbb{R}\times \mathbb{R} \times \cdots$, i.e, countably many copies of $\mathbb{R}$. $(-1,1)^{\mathbb{N}}$ is completely analagous. I don't want a ...
0
votes
0answers
16 views

Differentiable structure on product of manifolds to yield inclusion maps as imbeddings

I am working through Munkres' "Elementary Differential Topology" and trying to do every exercise, but this one question has me somewhat stuck. It is exercise c on page 11. The exercise is as follows: ...
0
votes
2answers
68 views

Banach-Alaoglu theorem, Rudin's functional analysis.

Few questions about the theorem If $V$ is a neighborhood of $0$ in a topological vector space $X$ and if $$ K = \left\{\lambda \in X^* : |\Lambda x | \leq 1 \; \text{for every} \; x \in V \right\}...
1
vote
1answer
55 views

Example of product topology where the index set is uncountable

I'm reading through Munkres, chapter 2, section 19 (Product topology). I can't see any example of product topology where the family of space is indexed with an uncountable set. Can you provide an ...
-2
votes
2answers
68 views

What is {empty set} x something with the product topology?

The product topology is defined as the topology induced by the basis of the product of open sets from each of the original topologies. From what I understand, then $\{\emptyset\}x]a,b[$ is an open ...
3
votes
2answers
144 views

Can every locally compact Hausdorff space be recognized as a subspace of a cube that has an open underlying set?

In this question cubes are topological spaces of the form $[0,1]^J$ with product topology and $[0,1]$ with usual topology. Further a space is a Tychonoff space if and only if it is a completely ...
0
votes
2answers
42 views

Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology.

Let X be a Banach space. Prove the mapping $(x,y)\mapsto x+y$ from $X \times X \to X$ is continuous when $X$ is given a weak topology. Could anybody give me some hints to start?
-1
votes
1answer
68 views

Help with product maps in topological spaces

Let $A, B, X, Y$ be topological spaces. Given two functions $f : A \to B$ and $g : X \to Y $, let $f \times g : A × X \to B \times Y$, $ (f \times g)(a, x) = (f(a), g(x))$. can you help me to show ...
1
vote
1answer
39 views

Set of continous functions writable as cartesian product

let $(X,d_X)$ and $(\mathbb{R}^n,d_2)$ be metric spaces with $X$ compact. Then the set of continuous functions is defined by $$ C_n(X):=\{f:X\rightarrow \mathbb{R}^n \;|\;f \text{ continuous }\} $$ $\...
1
vote
1answer
68 views

Topology closed sets in products of spaces

I got stuck at question 3.7.1 in Bert Mendelson's introduction to Topology. Prove that a subset $F$ of $X = \prod_{i=1}^nX_i$ is closed if and only if F is an intersection of sets, each of which is ...
1
vote
1answer
70 views

Formula for relative homotopy groups of products

In Hatcher prop 4.2 he proves that the n-th homotopy group of a product $X\times Y$ (for $X$ and $Y$ path-connected) is isomorphic to the product of the n-th homotopy groups of $X$ and $Y$. I wonder ...
4
votes
1answer
79 views

If $X \times X$ is normal, then is $X \times X \times X$ normal?

I am looking at some topological dimension theory for product spaces, and in trying to construct a certain type of counterexample it's become relevant to consider the question in the title above. I ...
1
vote
0answers
24 views

A product of intervals where the indexing set is all bounded sequences; Generalizing the limit of a sequence so that it is always defined.

This problem seems very hard to me, and any hints will be greatly appreciated. Here is the question: Let S be the space of bounded real sequences; for each $a\in S$ let $M_a$ be its least upper ...