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nLab > Latest Changes: Proper and improper subsets

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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeJul 29th 2010
    • (edited Jul 30th 2010)

    I found a good constructive definition of proper subset and put it in there. Also I wrote improper subset.

    Edit: also family of subsets; see below.

    • CommentRowNumber2.
    • CommentAuthorRodMcGuire
    • CommentTimeJul 30th 2010
    • (edited Jul 30th 2010)

    Given any way of expressing A as the intersection of a family of subsets of S, this family is inhabited.



    Umm, the empty family can in some way represent S which your definition avoids.
    However isn't S = intersection({S})?

    For an empty family to correspond to S you can't just use the definition of family of sets because it doesn't know about S. You have to define "family of subsets of a set S". If you demand that this definition not include S in the family then your definition works, but isn't this in someway hiding the notion of proper in the definition of "family of subsets of a set S"? If so then the definition should be of "family of proper subsets of a set S".
    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeJul 30th 2010

    However isn’t S = intersection({S})?

    True, but also S = intersection({}). Therefore there is a way of expressing S as the intersection of a family of subsets which is not inhabited, so S is not proper.

    Possibly your confusion is my fault, because I used the dangerous word ‘any’. I will change it to read ‘every’. That is, we are saying ∀F rather than ∃F.

    You have to define “family of subsets of a set S”.

    Yes, I assumed that the reader would know what that means. Perhaps we should link it and write family of subsets?

    For the record, a family of foos is (in general) a function from some set (called the index set of the family) to the set of all foos. In the case of a family of subsets of S, there is a trick that you can play with this definition if you want to be predicative and not assume the existence of the set of all subsets of S. That’s worth recording, so I will write family of subsets.

    you can’t just use the definition of family of sets

    Right, that’s completely different. It is really not a good idea to think of a subset of S as being a set; instead, a subset of S has a set associated with it. See subset and set for discussion of these issues.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeJul 30th 2010

    I have written family of subsets.

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