Due Mar. 13th in class
1. Suppose that f : X Y is continuous. If x is a limit point of the subset A of X, is it necessarily true
that f (x) is a limit point of f (A)?
Solution. In general, f (x) is NOT a limit point of f (A). He
Due Apr. 10th in class
1. (a) What are the components and path components of R (in product topology)?
(b) Consider R in the uniform topology. Show that x and y lie in the same component of R if and
only if the sequence
x y = (x1
Due Apr. 3rd in class
1. Let T and T be two topologies on X. If T T, what does connectedness of X in one topology imply
about the connectedness in the other?
Proof. The connectedness of X in T implies that of X in T. But not con
Due Mar. 20th in class
1. If each space X is Hausdor, then X is Hausdor space in both the box and product topologies.
Proof. Assume that X is Hausdor for all . For any given two points (x , (y X , (x ) = (y ,
then , there exist
1. Show that Q is countably innite.
Proof. We showed in class that Q>0 is countably innite. Now lets show that Q is countably innite.
First we denote by Q<0 the set of negative rational numbers. Then we dene a map
f : Q>0 Q<0
1. Show that if Y is a subspace of X, and A is a subset of Y , then the topology A inherits as a subspace
of Y is same as the topology it inherits as a subspace of X.
Proof. The space A as a subspace of Y has open subsets of
Due Mar. 27th in class
1. Let A X. If d is a metric for the topology of X, show that d|AA is a metric for the subspace topology
Proof. Lets check out the bases for these two topologies: the basis B for the subspace topolog
Due Feb. 20th in class
1. Let C be a collection of subsets of the set X. Suppose that and X are in C, and the nite unions
and arbitrary intersections of elements of C are in C. Show that the collection
T = cfw_X C|C C
is a topol