Definition talk:Torus (Topology)/General

The old definition was incorrect. It defined the topology as the set of all products, when it should have been the topology generated by the set of all products. But we already have tools for all of that, so we might as well use them. --CircuitCraft (talk) 20:01, 18 October 2023 (UTC)

You're very sure of yourself. Can you find a hard copy source to back this up? The work you're dissing here has been here since 2009 and the person posting that up was very sure of himself too. --prime mover (talk) 23:09, 18 October 2023 (UTC)

I apologize if I came off as condescending. I didn't substantially change the definition at all. Apart from changing the notation slightly (using $\prod$), I replaced what appeared to be a copy of the definition of the product topology with the already-defined version on this site. My intention was to leave the definition as-is, but make it more readable by using higher-level definitions.

The comment I had about it being incorrect was referring to what looks like a typo in the old definition of the topology. If interpreted as written, the resulting structure doesn't satisfy the open set axioms.

TL;DR I based this definition entirely off of the old one, just patching up some ambiguity/typos. I can go looking for a literature source using this formulation, but $\R^n / \Z^n$ may be easier to find. --CircuitCraft (talk) 05:32, 19 October 2023 (UTC)

I have reinstated the original definition as Definition 2. It remains to:

fix whatever typo you found in Definition 2

fill in the equivalence proof.

I trust you can do the needful? --prime mover (talk) 06:08, 19 October 2023 (UTC)