# Questions tagged [products]

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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### Continuity and Product Spaces

I have the following basic question regarding continuous functions. Is the following statement correct? Fix nonempty toplogical spaces $A, B, C, D$ with $A \times C$ and $B \times D$ endowed with the ...
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### For a basis of product topology on the $X \times X \times \cdots$，could there be infinitely many sets between two sets that are not the $X$?

By definition, a basis of the product topology on $$\prod_{\alpha \in J} X_\alpha$$ consists of sets of the form $$\prod_{\alpha \in J} U_\alpha,$$ where the $U_\alpha \neq X_\alpha$ only ...
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### Cut a unit stick at $n-1$ random points. Expectation of product of fragment lengths is $\prod\limits_{k=n}^{2n-1}\frac1k$. Why?

On a straight stick of length $1$, choose $n-1$ independent uniformly random points. Cut the stick at those points, yielding $n$ fragments. Let $\mathbb{E}_n$ be the expectation of the product of ...
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