What does it mean for a theorem to completely characterize something?

[fourier jr]

i've seen the phrase "this theorem completely characterizes such & such" in a couple books, but I'm still not really sure what it means. it's usually used before/after the proof of a (i assume) major theorem, and a definition usually comes out as a result of the theorem. that isn't really all there is to it, right?

 

[Tide]

How about giving an example? I've seen the expression used in the context of something like "the radius and location of the central point completely characterize a circle" which has a fairly clear meaning.

 

[fourier jr]

i guess it's just like that; maybe i just need to get used to the stuff I'm working on. today i went over the theorem that says if K is a compact Hausdorff space & f:X --> K is continuous then there is a continuous $$F: \beta X ->K$$ such that Foe = f where e is the embedding of X (evaluation map of f) in the cube $$\Pi I_g$$
In other words, every continuous function from X to a compact K can be extended to the Stone-Cech compactification of X. & then it goes on to say that this theorem "completely characterizes" the Stone-Cech compactification, up to topological equivalence (kind of like homeomorphism). The Stone-Cech compactification is the only compactification with this extension property (up to topological equivalence), & I guess that's why the book says the theorem characterizes it, right? I was just wondering because I see that phrase every so often & was never really sure what it meant.

 

[Tide]

For that, I'll have to defer to the mathematicians here because it's been more years than I care to admit since I took a point set topology course! :)

 

[matt grime]

In this case it means that if there is any other compact space with an embedding satisfying those rules you laid out then it is (probably canonically) isomorphic (in whatever the correct sense is) to the Stone Cech Compactification.

For instance, the direct product of two sets A and B is 'completely characterized' by being a set (which we label AxB) with projection maps onto A and B 'that are universal' where universal might need explaining. It means that if S is another set with projections onto A and B then there is map from S to AxB and the projections from S to A or B factor (uniquely) through the projections from AxB to A and B.

A non-unique characterization of something might be: an algebraic closure of a field is a field extension such that every polynomial splits. This is not unique. There is an algebraic closure of Q that is not the complex numbers, thus the closure of something is not uniquely characterised. There is probably a way to distinguish 'the smallest' closure though, just like there are many compactifications of a topological space that are *not* the Stone-Cech compactification. For instance, the Stone-Cech Compactification of the open interval (0,1) is (homeomorphic to) the circle S^1, yet there is also the [0,1] closure in R of it. I hope I've got that right: Stone-Cech puts one point at infinity for a connected set like that, doesn't it?)

A common thing that makes this distinction important is the idea that you are often making an infinite number of choices that need to be compatible (in your case, one for each point of the image set or something like that). Sometimes those choices turn out to be canonical and sometimes they are not.

The trivial group is completely characterised up to isomorphism by 'there is an injective group homomorphism from it to every other group'

proof: suppose there were another group with this property $G$, then there is an injection from $G$ to e, which is obviously a surjection, hence an isomorphism, moreover there is a map from e to any group $H$ sending e to e.

 

[Hurkyl]

No, the Stone-Cech compactification is the most general compactification:

If X is a space, C is a compact space, and the map X --> C has a dense image, then it factors uniquely through the Stone-Cech compactification.

 

[fourier jr]

there isn't really anything about that theorem in particular that i was having trouble with; it's just that it made me wonder what it meant for a theorem to characterize something, that's all.

 

[fourier jr]

No, the Stone-Cech compactification is the most general compactification:
If X is a space, C is a compact space, and the map X --> C has a dense image, then it factors uniquely through the Stone-Cech compactification.

yeah that's what part of this section in my book ends up showing, that the stone-cech compactification is the 'largest' compactification (& the 1-point compactification is the smallest, incidentally). i went over what i did yesterday 1 more time & went on to the next page & now i think i know what it means for a theorem to characterize something, at least in this case. the other examples given here helped out also. i guess i could have looked up characterize to find out what it means also: "to describe the character or quality of" & "to point out the chief quality or qualities of an individual or group".

 

[Hurkyl]

(& the 1-point compactification is the smallest, incidentally).

Only if the space is not already compact!