Let X be the collection of all nonempty subsets of the plane. For A and B in X, define <= to mean that B is a subset of A; Give a subset A of X such that A has an upperbound but does not have a supremum. But the intersection of A will always yield the least upper bound, unless this intersection is empty, in which case there are no upper bounds. [David Eger]
Proposition 0.25: [typo] It should read and suppose S is an initial segment of A not and suppose S is an intial segment of S as it reads in early prints. [David Eger]
Theorem 2.27: [Vacuous Logic] In the last line of the proof, I fail to see how T being a subset of Tb is a result of the fact that B is a subset of T. Usually, I am accustomed to seeing proofs of T being a subset of Tb being in the form t in T implies t in Tb. [David Eger]
Example 2.28 (f): [Vacuous Logic] This example makes a topology on the
integers generated by B(m,n) := {km+n, k integer}. Cain
wishes to show that any intersection of two elements of B is also an
element of B. In this case, as opposed to what he has
Suppose x is in B(m,n) intersect B(p,q). Then x is in B(mp,x) subset of
B(m,n) intersect B(p,q).
It should read:
Either B(m,n) intersect B(p,q) = {} as in the case of B(2,2) intersect
B(2,3), the even and odd integers respectively), OR there is some z so
that z is in B(m,n) and z is in B(p,q). In that case, B(m,n) intersect
B(p,q) = B( gcd(m,p), z).
[David Eger]
Example 3.8 (a): [Incorrect] It should read 0 <= a < b implies f^{-1}((a,b)) = (-sqrt{b}, -sqrt{a}) union (sqrt{a}, sqrt{b}) and a < 0 < b implies f^{-1}((a,b)) = (-sqrt{b}, sqrt{b}) [David Eger]
Example 3.8 (e): [Vacuous Logic] Cain is proving that if you have two real-valued continuous functions from a topological space X, then the function f-g is also continuous. But I do not follow the last few lines. He says that f(x0) - g(x0) in (a,b), and derives that in a neighborhood N of x, then g(x0) + a < f(x) < g(x0) + b, and f(x0) - b < g(x) < f(x0) - a, from which it is supposed to follow that a < f(x) - g(x) < b. I do not see it. [David Eger]
Theorem 3.11: [typo] At the end of the first paragraph of the proof, it should read f^{-1}(V) subset U follows from V = Y - f(X - int U) implies f^{-1}(V) subset int U subset U [David Eger]
Example 3.15 (b): [Incomplete Assumptions] Either f(X) needs to be both open and closed here, so either have f being onto, or make a note of where f(X) goes. [David Eger]
[Improper Ordering of Theorems] Proof of Thm 3.13 is properly moved to be the proof of Corollary 3.12, and Thm 3.13 is actually a corollary of 3.12 [David Eger]
Theorem 3.20: [Needed Closing to Proof] The very end of this proof is obvious but left unstated. For completeness I would note that C(x; r) is the union of all such B's, and from that it follows that C(x; r) member of w(X, F). But I suppose it is a matter of personal taste. [David Eger]
Example 10.3 (a):[typos] In the third case of f((a,b)) the interval should be closed at 0, not open. Also, a couple line downs in the long string of equalities, it should read f( union I) = union f(I) instead of f(intersection I) = intersection f(I). [David Eger]