1)  Let (X, T) be a topological space. Show that (X, T) is Hausdorff
if and only if the diagonal D= {(x, x): x is an element of X} is
closed in the product topology (X x X, T x T).

2) Suppose T1 and T2 are two topologies on X such that T1 is a subset
of T2. For a set A subset of X, let int1 (A) and clos1 (A) denote
respectively the interior and the closure of A in T1. Similarly
int2(A) and clos2(A) are defined. Establish inclusion relations
between int1(A) and int2(A), and clos1(A) and clos2(A). Construct
examples to show that those inclusions can be strict.


3)	On R, consider the topology T with a base consisting pf all subsets
of Q and all (usual) open intervals.

(a)	Give a simple description of open sets in this topology.
(b)	Is this topology Hausdorff?
(c)	Is this topology regular?
(d)	Is this topology connected?

4)	Let (X,d) be a compact metric space and f: (X,d) to (X,d) be an
isometry (i.e., d(f(x), f(y)) = d(x,y). Show that f is onto.

5)	State the contraction mapping principle. Using this show that each
f: Bn to Bn satisfying the condition  || f(x) ? f(y)|| is less than or
equal to ||x-y|| has a fixed point. Here, Bn is the closed unit ball
(w.r.t the usual norm) in Rn.

6)	Let X be the one-point compactification of (R, discrete). For each
of the following, write True or False with a simple justification.

(a)	X is compact.
(b)	X is Hausdorff
(c)	X is normal
(d)	X is completely regular
(e)	X is second countable
(f)	X is separable
(g)	X is connected.

7)	Let (X,d) be a complete metric space, f: X to X be continuous, A: X
to R be a nonnegative such that
D(x, f(x)) is less than or equal to A(x) ? A(f(x))
Show that f has a fixed point in X.

8) Suppose A is a subset of R ( which carries the usual topology). It
is given that every continuous function from A to R is bounded. Show
that A is compact.