Definitions
from The American Heritage® Dictionary of the English Language, 4th Edition
 n. A set contained within a set.
from Wiktionary, Creative Commons Attribution/ShareAlike License
 n. With respect to another set, a set such that each of its elements is also an element of the other set.
 n. A group of things or people, all of which are in a specified larger group.
from The Century Dictionary and Cyclopedia
 n. In mathematics, a proper part or the whole of the set: thus, a subset of S is a set every element of which belongs to S.
from WordNet 3.0 Copyright 2006 by Princeton University. All rights reserved.
 n. a set whose members are members of another set; a set contained within another set
Etymologies
Examples

Indeed, the bigger point to be made is that empiricism is all that we have to explain anything that we wish to understand, “physics” being only what we call a subset of empiricallybased theories.

Re: my use of spanish phrases, it pisses off a certain subset of conservatives, just like my "People Eating Tasty Animals" hat pisses off a certain subset of liberals.
Tom McIntyre Explains His Picks for our 2009 Hunting and Fishing Heroes and Villians FaceOff

Yeah, I'm not sure how high parental involvement is ever going to be outside of a certain subset of small uppermiddleclass schools.
intertribal: things about public education I've learned from this job

You can call it what you want, but an ordered set such that each subset is contained within its superset is called a nested hierarchy.

In order for thenre to be an ordered set such that each subset is contained within its superset, all members must share the same defining characteristics.

This forms an ordered set such that each subset is strictly contained within its superset.

What would you call an arrangement such that each subset is strictly contained within its superset?

Each subset is strictly contained within its superset.

Each subset is strictly contained within its superset, hence is meets the definition of a nested hierarchy to which you agree above.

For the purposes of discussion, what should we call an ordered set such that each subset is strictly contained within its superset?
thtownse commented on the word subset
Bilby, I hope you get over your code.
October 19, 2010
bilby commented on the word subset
I like to sit and watch the subset by the ⊂.
October 19, 2010
nuxiy commented on the word subset
A set in a set *thumbsup*
May 20, 2009