Talk:Subspace of Subspace is Subspace

"Ambient space" is just a word to describe the result, much like "depend".

Still needs to be defined. As it stands, the title is too far different from the exposition for a novice to be able to make sense of it. --prime mover (talk) 16:59, 30 August 2017 (EDT)

Maybe. If we change the title, I'd certainly use "topology" and "depend". What about "Subspace Topology does not depend on Superset"? --barto (talk) 17:22, 30 August 2017 (EDT)

Moving a bit off-topic: Any attempt to define "ambient space" will probably result in a disambiguation with about 20 links to pages that are informal explanations rather than definitions. --barto (talk) 17:22, 30 August 2017 (EDT)

There exist the following options:

a) Rewrite the title so as not to use the term "ambient space".

b) Define the concept "ambient space". (It's a concept, it is not defined on $\mathsf{Pr} \infty \mathsf{fWiki}$, and if it is going to be used, it needs to be defined.) If it results in a "disambiguation with about 20 links to pages ..." that means either:

it cannot be defined accurately and precisely, and so it is not a mathematical term and so we don't use the word here

or:

it needs someone with better explanatory skill to define it.

What we do not do is leave technical terms undefined. --prime mover (talk) 18:25, 30 August 2017 (EDT)

As you can see this is not covered by Topological Subspace is Topological Space. --barto (talk) 16:52, 30 August 2017 (EDT)

No, not specifically, but there do exist a number of results that do the same sort of job. I believe they may cover it. --prime mover (talk) 16:59, 30 August 2017 (EDT)

I have been unable to find anything that already does this. If something does already exist, let me know and I’ll sort out the duplication. —Leigh.Samphier (talk) 07:43, 27 June 2019 (EDT)

Okay no worries. What's here is good. If we find duplication, we can place a mergeto template in there and it can then be covered as and when we get to it. --prime mover (talk) 11:13, 27 June 2019 (EDT)

A few weeks back I noticed that there were similar theorems that I could not find:

Subring of Subring is Subring

Subgroup of Subgroup is Subgroup

Restriction of Restriction of Operation is Restriction

Restriction of Restriction of Function is Restriction

Restriction of Restriction of Relation is Restriction

I wasn’t sure if I was being overly pedantic/pedestrian about this. I think these theorems are deemed so obvious that they are not stated. You rarely find them in a book. But it is common to move between a structure and it’s substructure in theorems and the substructure definition is designed to allow this. —Leigh.Samphier (talk) 17:25, 27 June 2019 (EDT)

If you can find them in a book then someone deems them worthy of stating. I can't see a need for such depth of detail myself.

As a subgroup, for example, is defined as being a subset of a group which is itself a group, then it means invoking "Subgroup is subset of group which is group, subgroup of subgroup is subset of subset of group which is group, the result follows from subset transitivity" -- and unless you are on shaky conceptual ground where you need to be very careful about whether or not you are "allowed" to use transitivity of subsets, it's probably a detail which can be taken for granted.

And the same applies to all such statements. But if you feel like posting them up, feel free. --prime mover (talk) 18:02, 27 June 2019 (EDT)