WA 5: Solutions to some problems
2.1.2. If the operator T : X Y is continuous, then the pre-image of any closed
set in Y is closed in X , so the pre-image of cfw_0, that is, ker T is closed.
Conversely, if ker T is closed, then the quotient map Q : X X /
WA 3: Solutions to some problems
I didnt include part 2 of Problem 1.4.19 and part 2 of Problem 1.4.23 in your
grade (they are too advanced for this class and require an additional knowledge of
topology).
Problem 1.4.19 (part 1). We will rst show that any
WA 2: Solutions
Problem 1.2.6. First of all, observe that X has no largest element. Indeed,
otherwise for such an element x the set mincfw_x cfw_x = X would be countable. Let
y X be such that x < y . Then if x is not the rst element of X , consider the se
WA 1: Solutions to some problems
1.1.3. Dene an order on the system S of order isomorphisms : X Y ,
where X and Y are segments of X and Y , respectively, as follows: means
that X X and |X = (it is easy to verify that this is indeed an order).
Then the sys