# Tagged Questions

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### How to find subbase and base for $X\times Y$?

Let $\tau :=\{X,\emptyset,\{a\},\{b,c\}\}$ on $X=\{a,b,c\}$ and $\tau^*:=\{Y,\emptyset,\{u\}\}$ on $Y:=\{u,v\}$ i) Find a subbase for the product topology on $X\times Y$ ii) Find a ...
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### Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
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### A product of two sequentially compact metric spaces is compact. How to prove this explicitly?

We know that a product of two (or finitely many) compact topological spaces is compact. And we also know that in a metric space, compactness is equivalent to sequential compactness. So a product of ...
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### Is this a homeomorphism?

Suppose you have a cartesian product of spaces $\prod_{\alpha\in\mathcal{A}}X_{\alpha}$ in the product topology. Choose any $\alpha\in\mathcal{A}$ . Is the following a homeomorphism of a subspace ...
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### Homeomorphism of product of topological spaces

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 ...
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### Definition of product of uniform spaces

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous. But Springer's encyclopedia ...
Let $X,Y,Z$ be topological spaces. It is well-known that if $F:X\times Y\to Z$ is a continuous map, we can define a map $$\overline{F}:X\to C(Y,Z) \\\overline{F}(x)(y)=F(x,y)$$ where $C(Y,Z)$ is the ...
### Is $\prod_{\mathbb{R}}\mathbb{R} = \mathbb{R}^\mathbb{R}$?
(If the title is unclear, I'm looking at infinite cartesian product of $\mathbb{R}$ indexed by $\mathbb{R}$.) I thought that I had reasoned this rather well, as follows: \$\mathbb{R}^\mathbb{R} = ...