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# 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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### 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, ...
52 views

### $X\times Y$ 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 ...
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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. ...
1 vote
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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 ...
56 views

### 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 ...
1 vote
55 views

### 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 ...
1 vote
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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 ...
138 views

### 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 $F$ ...
148 views

### 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 ...
55 views

### 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 ...
51 views

### 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-...
50 views

### 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 ...
1 vote
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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 $g(x) = \inf_{y \in Y} f(x,y)$. ...
62 views

### 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 ...
169 views

### 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 ...
47 views

### 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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### 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 \$...