pg. 3: if A is closed, bar{A}=A.
Add note to end of 1-3: in a Hausdorff space, a sequence can only converge to 1 point.
pg. 4: Delete the 2nd "i.e...." statement near the end of 1-5.
In the definition for completely regular space, f(A)={1}.
In the definition for normal, "exists" should be "exist."
pg. 5: 2nd line, "converge" should be "converges.
Add: The following are normal.
*Regular space with countable basis
*Metrizable space
*Compact Hausdorff space
Add: if Y is compact in Hausdorff X, x_0\nin Y, there are disjoint open sets U,V of X containing x_0,Y.
Add: A countable product of 1st-countable spaces is 1st-countable.
pg. 6: Delete last sentence of 1-10.
Add: If H is normal then G/H is a topological group.
pg. 7: Arrowa A->B means B is subset of A.
Add the following arrows:
compactly generated -> 1st axiom countability
1st axiom countability -> 2nd axiom countability
Lindelof -> compact
Baire space -> complete metric space
Connected -> path connected
Delete the following arrows:
Locally connected -> connected
Locally path connected -> path connected
Baire space -> metric space
Delete "a finite subcover" from countably compact. Centered means the intersection of a finite number of sets in the system is nonempty.
Delete "countable" from compact.
For completely regular space, replace "f(A)={0}" with "f(B)=1".
Add to connected: "Equivalently, no separation (open proper subsets A and B such that AUB=X.)
Add "Hausdorff" to hypothesis of Smirnov.
Add: In a metric space, Lindelof=separable=2nd countable
pg. 8 Delete last statement in 1st paragraph.
Extreme Value Theorem: change "f is compact" to "f is continuous"
pg. 9
compact subspace C of x (not X).
In existence part of proof,
*Delete "C" at end of period.
*add X-C (not add C)
*Replace "Y is compact" with "Y is Hausdorff"
Converse is proof that X is locally compact.
For part 2 of Stone-Cech, note: If h:X->Z is an imbedding where Z is compact Hausdorff, there is a compactiication Y of X as h extends to an imbedding H:Y->, unique up to equivalence.
pg.10
2-5 note open refinement is defined in 3-6.
pg. 11
Last statement of 3-1: switch d, d'
3-2 Add the following:
Lebesgue number lemma: If A is an open cover of (X,d) and X is compact, then there exists delta>0 such that for each subset of X with diameter Finer.
When J is infinite, R^J is not metrizable in product or box topology.
p.12
In Urysohn, replace regular space with normal space.
p.13 Let X be a regular space with countably locally finite base B.
p.15 Item 2: both open and closed.
Delete "Cartesian" in item 5 of basic results.
IVT: Replace "c is between" with "r is between"
p.16
Proof of existence: replace \phi_i with \phi_j.