1. ## Homeomorphic spaces

I came up with this nice little theorem, can anyone verify or deny it?

I can show my (very very long...need like six lemmas) proof, but it goes like this:
Theorem:: Let $\displaystyle \left\{X_j\right\}_{j\in\mathcal{J}}$ and $\displaystyle \left\{Y_j\right\}_{j\in\mathcal{J}}$ be two collections of topological spaces such that $\displaystyle X_j\approx Y_j$ for all $\displaystyle j\in\mathcal{J}$ ($\displaystyle \approx$ means homeomorphic). Then, $\displaystyle \prod_{j\in\mathcal{J}}X_j\approx\prod_{j\in\mathc al{J}}Y_j$ under the product topology.

2. Originally Posted by Drexel28
I came up with this nice little theorem, can anyone verify or deny it?

I can show my (very very long...need like six lemmas) proof, but it goes like this:
Theorem:: Let $\displaystyle \left\{X_j\right\}_{j\in\mathcal{J}}$ and $\displaystyle \left\{Y_j\right\}_{j\in\mathcal{J}}$ be two collections of topological spaces such that $\displaystyle X_j\approx Y_j$ for all $\displaystyle j\in\mathcal{J}$ ($\displaystyle \approx$ means homeomorphic). Then, $\displaystyle \prod_{j\in\mathcal{J}}X_j\approx\prod_{j\in\mathc al{J}}Y_j$ under the product topology.

It turns out this is correct.

If anyone wants to see a proof, take a loot at my blog at the bottom of the page.