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Rasalhague
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Wikipedia: Neighbourhood:
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?
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,
[tex]p \in U \subseteq V[/tex]
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?