# 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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### 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 ...
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### 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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### 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 $$\Delta:=\{(x,y)\in \mathcal{X}^2:x=y\}$$ ...
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### 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 $\Bbb R^n$?
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### 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 ...
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### 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 $U \times V$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
### If $X \times X$ is normal, then is $X \times X \times X$ normal?
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 ...