Containment Problem:Neighbourhood Definition

[Rasalhague]

Wikipedia: Neighbourhood:

If X is a topological space and p is a point in X, a neighbourhood of p is a set V, which contains an open set U containing p,

$$p \in U \subseteq V$$

It is standard practice to use the word "contains" in this intuitive, but potentially ambiguous way, to mean either "is an element of" or "is a superset of"? I've seen it used like this on a few Wikipedia pages.

Carol Whitehead, in her Guide to Abstract Algebra, uses "contains" in the latter sense (§ 1.1, p. 9, Notation), as a gloss for "is a superset of". I remember this distinction as one of the things that most confused my intuition when I first read about sets, in particular when I first met the idea of the empty set, and so, for that reason, I try to be careful about it in my notes.

On the other hand, as I mentioned in a thread here a while ago, I can see why it could be handy to have an intentionally loose term meaning either. Even when people define a vector space as a tuple, such as ((set of vectors, addition of vectors), (set of scalars, addition of scalars, multiplication of scalars), multiplication of vectors with scalars), which itself is defined, more fundamentally, as a complicated hierarchy of nested sets, this rarely inhibits them from talking about a vector being in (or even an element of) this vector space. It would be nice if there was a nonspecific expression (such as "in" or "contained by") that could be used in such cases without loss of rigor, and that would gel with our intuition of containment as transitive; and that was kept distinct from the technical terms "an element of" and "a subset of".

But how does this match with actual current usage?

 

[micromass]

Not sure what your getting at here.

I see no problems with the word "containment". Of course, using words to describe a situation is non-rigurous. What you should actually do is replace all words with symbols. That's how the math was actually meant to exist. But it takes us hours just to figure out 1 sentence of symbols, so we write in words.

If you read a proof of a theorem in a book, then you should be aware that the proof isn't the written words. It's only when you convert the words to symbols that you could obtain the real proof. I would like to check out the website http://us.metamath.org/mpegif/mmset.html In this website, all theorems are provided with their real proofs. That is, no words involved, only the symbols. (needless to say that only a few will understand any of it )


As for the vector space notation. I have to agree: saying that you take a vector in a vector space is incorrect. It would be better to say that you take a vector in the underlying set of the vector space. But then again, this would complicate many sentences and many proofs, so we don't really do that. You should only do that if you want the real proof (aka the proof with no words). But outside of it, I see no reason to be overly rigorous...

 

[Rasalhague]

Hey, thanks for the link, micromass. That looks really interesting. It's a relief to know that it can be done, even if it may take a while to get the hang of it!

Since the actual mathematics, the ideas, is being communicated in largely words, it should be (and is) possibly to translate unambiguously between words and notation, albeit sometimes at the cost of brevity. But a clever and consistent choice of words (and of notation) can make it easier to translate between them, and cut down on misunderstandings. Funnily enough, I recently came across the argument that the important thing is the words, and the notation is just abbreviations for statements in natural language. But I suppose they're both systems of symbols (in the broader sense).

A potential problem I can see with using "in" and "is contained by" in this loose way is that the difference between$\in$ (LaTeX "\in"!) and $\subseteq$ is a little bit unituitive at first, as is the fact that they aren't transitive (hereditary). When mathematical ideas at at odds with untrained human intuition, I think it's helpful to have that difference made explicit, and to drum it in with consistent use of terms. So many times I've been thrown into confusion by "convenient notations" (symbolic or verbal), that sacrifice transparency for tradition, and don't make clear how seriously their appeal to intuition is to be taken...

As for the vector space notation. I have to agree: saying that you take a vector in a vector space is incorrect. It would be better to say that you take a vector in the underlying set of the vector space. But then again, this would complicate many sentences and many proofs, so we don't really do that.

This is not too confusing, just a bit niggly. Maybe we could make a definition: [insert convenient term here] will mean "an element of, or an element of an element of, or... (and so on)" OR "a subset of, or a subset of a subset of, or..." OR "an element of a subset of a... (and so on, with any permutation of 'element of' and 'subset of')", and invent some symbol to express the idea. Then we can have rigor and convenience?

 

[Rasalhague]

The idea behind saying a vector is "in a vector space, V" must be that the set V gains properties from its setting/context: the set V becomes a vector space by being (in the required relationship) "with" certain other sets. But this natural way of thinking seems to clash with the way the ideas been formalized in terms of these entities called sets which are supposed to have only minimal properties. A space is sometimes called a set with some extra structure, but sets are defined not to have any more the most basic structure: what their elements are.

On the other hand, thinking further about this... we say a set, A, "has" a cardinality, and it has the property of being a subset or a superset of other sets, and sets can be categorized according to the properties of their elements. We don't think of properties such as a set's cardinality, or its property of being a subset of other sets X, Y, Z..., or its property of being an element of sets P, Q, R..., as being themselves "in" the set A, even though A "possesses" these properties, and so the properies might be said, informally, to "belong to A".

Defining a vector space, for example, as ((V,+),(F,+,*),f(F,V)) is like defining a "set with the property E" (E being, for example, the property of being an element of some other set, K, of a certain, specific type) as the pair (S,K), and then insisting that it's formally incorrect to refer to (what everyone calls) elements of such a set as "an element of this set-with-property-E, and instead considering the only correct expression to be "an element of the underlying set of this set-with-property-E" (and yet, not actually ever using this "correct" expression, for obvious reasons!).

So maybe a more natural way to formalize the way people actually think and talk about spaces would be to define the underlying set V as the vector space itself, and to formalize the property of being a vector space as the property of being the underlying set of the vector space structure tuple ((V,+),(F,+,*),f(F,V)). Similarly for other spaces (sets with extra structure), topological spaces, metric spaces, affine spaces, manifolds etc.

 

[Landau]

You expresses similar feelings in this thread.

Everyone agrees there is potential confusion, and knows that he/she is actually being sloppy when saying thing like 'a vector in a vector space'. But after a while you get used to it and the confusion is minimal. Introducing new words for these things will likely confuse even more.

I can't remember ever being confused about the use of 'containment'. But yes:

"a set V, which contains an open set U containing p"

could a priori mean either mean that U is a subset of V, or that U is an element of V. The second meaning is just not sensible in this context: V is a subset of the space X (granted, this could have been explicitly included), so it consists of points in the space. The set U being open means (by definition of 'open'), that U is also a subset of X. So saying that V contains U cannot mean anything other than that U is a subset of V.

 

[Rasalhague]

Yeah, just the niggly part of my brain getting hyperactive again ;-)

At least there seems to be a consensus over which is the correct wording. I wouldn't want to change that.

I recall my delight at first reading about the axioms of groups, fields, vector spaces etc., at having it all layed out clearly and unambiguously right down to the building blocks, and then turning the page to read: "A vector space V is a 3-tuple (V,F,f), consisting of an abelian group V, [...] can be combined with every element u in V" (three different senses given to the same symbol in one paragraph). I'm more used to it now, but when all these concepts were new to me, I'd carefully anotate what I read, to make sure I knew which kind of V they were talking about on which occasion.