1
vote
2answers
85 views

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 ...
0
votes
1answer
243 views

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 ...
2
votes
2answers
99 views

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 ...
1
vote
1answer
241 views

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 ...
0
votes
1answer
170 views

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 ...
5
votes
1answer
168 views

Continuous maps from products of topological spaces

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 ...
3
votes
1answer
231 views

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} = ...