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

Wednesday, 23 January 2013 04:15Web

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 subspac

**e.**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 spac

**e.**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. |