Indian Statistical Institute (ISI) 2006-1st Year M.Math General Topology - Question Paper

Indian Statistical Institute
M.Math I Year
1st Semester Back Paper Examination, 2005-2006
General Topology
Time: three hrs
Attempt all ques.. All ques. carry equal marks. Any outcome proved
in the class may be cited and used without proof.

1. Let X be a topological space such that every real valued function on
X is continuous. Determine the topology on X.

2. Let F IRn be a closed subspace. Prove or disprove: F is connected if
and only if F is path connected (give a proof if actual or a counterexample
if false).

3. Let X, Y be topological spaces, f : X Y be a continuous map
having a continuous part s : Y X i.e., f s = 1Y . Prove that f
is a quotient map.

4. a) Let X be a topological space. Prove that every path connected
subspace of X is contained in a unique path component of X .
b) Let p0(X ) denote the set of path components of X . For f : X Y
continuous, let p0(f ) : p0(X ) p0(Y ) be the function that maps
a path component C of X to the unique path component of Y that
contains f (C). Let g : X Y be continuous. Show that if f g then
p0(f ) = p0(g).

5. Prove or disprove: S1 × S1 × S1 is homotopically equivalent to S2 × S1.
(give a proof if true, a counterexample if false).

Earning:   Approval pending.