LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
|
M.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – APRIL 2008
MT 3803 / 3800 – TOPOLOGY
Date : 26/04/2008 Dept. No. Max. : 100 Marks
Time : 9:00 – 12:00
Answer ALL questions.: (5 X 20 = 100)
- (a) (i) Let X be a metric space with metric d.
Show that d1 defined by
is also a metric on X.
(OR)
(ii) In any metric space X, prove that each open sphere is an open set.
(b) (i) Let X be a complete metric space and let Y be a subspace of X. Prove that
Y is complete iff Y is closed.
(ii) State and prove Cantor Intersection Theorem. (8+7)
(OR)
(iii) Let X and Y be metric spaces and let f be a mapping of X into Y.
Prove that f is continuous at x0 iff and f is
continuous iff f-1(G) is open in X whenever G is open in Y. (15)
- (a) (i) Prove that every second countable space is reparable.
(OR)
(ii) Prove that every separable metric space is second (5)
(b) (i) Show that every continuous image of a compact space is compact.
(ii) Prove that any closed subspace of a compact space is compact. (8+7)
(OR)
(c) (i) If f and g are continuous real or complex functions defined on a metrical
space X, then show that f+g and are also continuous.
(ii) Let C(X,R) be the set of all bounded continuous real functions defined on
metric space, show that C(X,R) is a closed subset of the metric space. (8+7)
- (a) (i) State and prove Tychnoff’s Theorem.
(OR)
(ii) Show that every sequentially compact metric space is compact. (5)
(b) (i) Show that a metric space is sequentially compact it has the Bolzano
Weierstrass property.
(ii) Prove that every Compact metric space has the Bolzano Weierstrass
Property. (10+5)
(OR)
(iii) State and prove Lebesgue Coverity Lemme.
(iv) Prove that a metric space is compact it is complete and totally
bounded. (9+6)
- (a) (i) Show that every subspace of a Hausdorff space is also Hausdorff.
(OR)
(ii) Prove that every compact Hausdorff Space is normed. (5)
(b) (i) Prove that the product of any non-empty class of Hausdorff Spaces is a
Hausdorff Space.
(ii) Prove that every compact subspace of a Hausdorff space is closed.
(iii) Show that a one-to-one continuous mapping of a compact space onto a
Hausdorff Space is a homeomorphism. (6+4+5)
(OR)
(iv) State and prove Uryshon Imbedding Theorem. (15)
- (a) (i) Show that any continuous image of a connected space is connected.
(OR)
Prove that if a subspace of a real line is connected, then it is an internal. (5)
(b) (i) Show that the product of any non-empty class of connected spaces is
connected.
(ii) Let X be a Compact Hausdorff Space. Show that X is totally disconnected,
iff it has open base whose sets are also closed. (6+9)
(OR)
(iii) State and prove Weierstrass Approximation Theorem.
Latest Govt Job & Exam Updates: