# 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.

304 questions
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### Compactness and countably compactness in metric spaces

I have a little doubt about compactness in metric spaces. I have this homework where I have to prove that $[0,1]^\omega$ with the uniform topology is not countably compact. As a consequence of ...
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### Countability of Topologies and Product Topology

I was trying to examine examples of topologies as a self learning project (you can see the list here), and these questions arose: I couldn't spot a product topology where any set other than basis ...
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### Non-homeomorphic structures and the Descartes' theorem

Some structures like the donut are not homeomorphic to a sphere. According to this link (https://en.wikipedia.org/wiki/Angular_defect#Positive_defects_on_non-convex_figures) the basis of the Descartes'...
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### Continuous function mapping non-open sets to open sets

While studying the product topology, I've come across an example that I can't shake. Consider the product topology on $\mathbb{R}^2$. We know that 1) This product topology is the same as the "...
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### Embedding of rational number Q into real number R is dense and the embedding of Q into p-adic numbers $Q_p$ is dense too

Q: If $p,q$ are different primes, show the embedding $$\mathbb{Q} \rightarrow \mathbb{Q}_q \times \mathbb{Q}_p$$ $$x \rightarrow (x,x)$$ is dense in the product space $\mathbb{Q}_q \times \mathbb{Q}_p$...
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### Is the sequence of real number that are $0$ from some point Sense in the box or product topologies

Let $A=\{(x_n)_{n\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}|\exists M\in\mathbb{N} ,\forall n>M, x_n=0 \}\subset\mathbb{R}^\mathbb{N}$, series of real numbers that are zero from some point forward. ...
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### Ultrafilters over product spaces

Suppose that for $i\in I$, $X_i$ are topological spaces and $U_i$ is an ultrafilter over $X_i$. Consider the space $\Pi_i X_i$ with the product topology. I want to know when, if ever, it is possible ...
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### Compact subspaces of the sigma product

Let $\prod_{\alpha<omega_1}(I)$ be the Tychnoff product of the unit interval $I$, then the product topology is Hausdorff compact topology. We know that the sigma product $\Sigma(0)$ as a subspace ...
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### What do opens in the product topology over $\mathbb{Z}_p$ look like?

Let $p$ be a prime, let $\pi_n : \mathbb{Z}/p^{n+1}\mathbb{Z} \to \mathbb{Z}/p^n\mathbb{Z}$ be the obvious projection and \mathbb{Z}_p = \{\, x \in \Pi_{n \geq 1} \mathbb{Z}/p^n\mathbb{Z} \,|\, \...