@John

>\\(f_*(A)\\) gives you the buckets all of whose balls are from \\(A\\). If a bucket has no balls in it, it's vacuously true that all the balls in this bucket are from \\(A\\).

Thanks! This pretty much cleared up all confusion I was having for a week. Basically its like elementary school field day. Everyone (\\(all\\)) gets an award for participation!

>These are quite different. For example, suppose a bucket has two balls in it: one from \\(A\\) and one not from \\(A\\). Then this bucket is in \\(f_{*}(A)\\) but not in \\(f_{!}(A)\\). _Some_ ball in that bucket is from \\(A\\), but not _all_.

>Or suppose a bucket has no balls in it. Then this bucket is in \\(f_{!}(A)\\) but not in \\(f_{\ast}(A)\\). _All_ balls in that bucket are from \\(A\\), but not _some_. If this seems surprising, read my previous comment. Since there are no balls in this bucket, it's *vacuously* true that all balls in this bucket come from \\(A\\).

I think you switched \\(f_!\\) and \\(f_*\\) here?

>\\(f_*(A)\\) gives you the buckets all of whose balls are from \\(A\\). If a bucket has no balls in it, it's vacuously true that all the balls in this bucket are from \\(A\\).

Thanks! This pretty much cleared up all confusion I was having for a week. Basically its like elementary school field day. Everyone (\\(all\\)) gets an award for participation!

>These are quite different. For example, suppose a bucket has two balls in it: one from \\(A\\) and one not from \\(A\\). Then this bucket is in \\(f_{*}(A)\\) but not in \\(f_{!}(A)\\). _Some_ ball in that bucket is from \\(A\\), but not _all_.

>Or suppose a bucket has no balls in it. Then this bucket is in \\(f_{!}(A)\\) but not in \\(f_{\ast}(A)\\). _All_ balls in that bucket are from \\(A\\), but not _some_. If this seems surprising, read my previous comment. Since there are no balls in this bucket, it's *vacuously* true that all balls in this bucket come from \\(A\\).

I think you switched \\(f_!\\) and \\(f_*\\) here?