LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
SUPPLEMENTARY EXAMINATION – JUNE 2007
MT 3803 - TOPOLOGY
Date & Time: 26/06/2007 / 1:00 - 4:00
Dept. No.
Max. : 100 Marks
Answer all the questions.
01.(a)(i) Let X be a non–empty set, and let d be a real function of ordered pairs of
elements of X which satisfies the following two conditions:
d(x,y) = 0  x = y, and d(x,y)  d(x,z) + d(y,z)
Show that d is a metric on X.
(OR)
(ii) Let X and Y be metric spaces and f be a mapping of X into Y.
Prove that f is continuous at xo if and only if xn x0  f(xn)  f(x0).
(b)(i) Let X be a complete metric space, and let Y be a subspace of X.
Show that Y is complete  it is closed.
(ii)
Let X be a complete metric space, and let {Fn} be a decreasing sequence of
non–empty closed subsets of X such that d(Fn)  0.

Prove that F =  Fn contains exactly one point.
n 1
(iii)
State and prove Baire’s Theorem.
(6+5+4)
(OR)
(iv) Prove that, the set C(X, R) of all bounded continuous real functions
defined on a metric space X, is a real Banach space with respect to
pointwise addition and scalar multiplication and the norm defined by
f = sup f (x) .
02.(a)(i)
If a metric space X is second countable, show that it is separable.
(OR)
(ii) Let X be a any non–empty set, and let S be an arbitrary class of subsets
of X.
Prove that the class of all unions of finite intersections of sets in S is a
topology.
(b)(i) Show that any closed subspace of a compact space is compact
(ii) Give an example to show that a proper subspace of a compact space need
not be closed.
(iii) Prove that any continuous image of a compact space is compact.(5+4+6)
(OR)
(iv) A topological space is compact, if every subbasic open cover has a finite
subcover.
Prove that a metric space is sequentially compact  it has the
Bolzano–Weierrtrass property.
(OR)
(ii) Show that a closed subspace of a complete metric space is compact  it is
totally bounded.
03.(a)(i)
(b) (i) State and prove Lebesgue’s covering Lemma.
(ii) Show that every sequentially compact metric space is complete.
(9+6)
(OR)
(iii) Prove that the product of any non–empty class of compact spaces is
compact.
(iv) Show that any continuous mapping of a compact metric space into a
compact metric space is uniformly continuous.
(7+8)
04.(a)(i)
(ii)
In a Hausdorff space, show that any point and disjoint compact subspace
can be separated by open sets.
(OR)
Prove that every compact Hausdorff space is normal.
(b)(i) State and prove the Tietze Extension Theorem.
(OR)
(ii) If X is a second countable normal space, show that there exists a
homeomorphism f of X onto a subspace of R∞, and X is therefore
metrizeble.
05.(a)(i)
(ii)
Show that any continuous image of a connected space is connected.
(OR)
Let X be a compact Hausdorff space, show that X is totally
disconnected  it has an open base whose sets are also closed.
(b)(i) Let X be a topological space and A be a connected subspace of X.
If B is a subspace of X such that A  B  A , then show that B is
connected.
(ii) If X is an arbitrary topological space, then prove the following:
(1) each point in X is contained in exactly one component of X;
(2) each connected subspace of x is contained in a component of X;
(3) a connected subspace of X which is both open and closed is a
component of X.
(6+9)
(OR)
(iii) State and prove the Weierstrass Approximation theorem.
X
X
X