Indian Statistical Institute (ISI) 2005 M.Sc Mathematics General Topology - Question Paper

Attempt any 5 ques.. All ques. carry equal marks. Any outcome
proved in the class may be cited and used without proof.

1. a) Let X be compact and Hausdorff, A ( X be closed. Show that X/A
is homeomorphic to the one-point compactification of X - A.

b) define explicitly the quotient topology on the quotient group
IR/|Q, IR being the real line, |Q the set of rationals, treated as a subgroup
of the group (IR, +).

2. a) Prove that GL(n, C) is path connected (hint; use the polynomial
p(z) = det((1 - z)I + zA) for A two GL(n, C)).

b) Prove that any discrete subgroup of S1 must necessarily be finite
cyclic.

3. a) Let X be any space. Show that CX, the cone over X is contractible.

b) Show that Sn-1 is a deformation retract of Sn - {N, S},N and S being the north and south poles of Sn respectively.

4. Let f, g : X ! Sn be continuous maps with f(x) 6= -g(x) eight x two X.
Prove that f ' g.

5. Let X be a space. Then show that X is path connected if and only if
all constant maps: X ! X are homotopic to every other.

6. Let R_ : S1 ! S1 be a rotation by angle _. Show that R_ is homotopic
to the identity map: S1 ! S1.

7. Let G be a connected group, H a discrete normal subgroup. Prove that
H _ Z(G), the centre of G.

Earning:   Approval pending.