Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
PGMT-6A (PT/10/VIA) PGMT-6A (PT/10/VIA) POST-GRADUATE COURSE e) Term End Examination — December, 2014 / June, 2015 Paper - 6A : General Topology Full Marks : 50 ( Weightage of Marks : 80% ) Special credit will be given for accuracy and relevance in the answer. Marks will be deducted for incorrect spelling, untidy work and illegible handwriting. The marks for each question has been indicated in the margin. Answer Question No. 1 and any four from the rest. 1. Answer any five questions : a) b) 2. f) Prove that the set of all irrational numbers with the usual topology induced from the usual topology of reals is totally disconnected. g) Construct a uniformly v for the space R of reals which induces the usual topology of R. a) State the Kuratowski closure axioms. Prove that the involved operator c ( say ): (X ) (X ) generates a topology ( say ) 2 5 = 10 1 1 1 If A 1, , ,..., , obtain limit points n 2 3 of A, if any, of A if A is endowed with (i) usual topology of reals, (ii) discrete topology, (iii) cofinite topology. on X such that c ( A ) closure of A in ( X , ) , for b) d) Show that a finite topological space that is T1 has discrete topology. PG-Sc.-1310-G [ P.T.O. where X denotes the 4 In a topological space ( X , ) if G is an open and only if G A . ( both closed and open). suitable topological spaces so that f is both open and closed but not continuous. AX, set and A X then show that G A if In a topological space ( X , ) if A X , show Give an example of a mapping f : ( X , ) (Y , ) where ( X , ) , (Y , ) are all underlying set. that Bdr (A ) if and only if A is clopen c) Prove that in a topological space ( X , ) the set A consisting of the elements of a convergent sequence along with its limit point is compact. MATHEMATICS Time : 2 Hours 2 3. 3 c) Prove that the real number space R with the lower limit topology is separable but is not second countable. 3 a) Let f : ( X , ) (Y , ) . Prove that following statements are equivalent : i) f is continuous. ii) For any closed set F in Y, f closed in X. PG-Sc.-1310-G 1 (F ) is 3 PGMT-6A (PT/10/VIA) f (A ) f (A ) for every subset A of X. iv) f 1 (B ) f 1 (B ) for every subject B of Y. b) b) iii) of A X if and only if there is a net in A \ { u } such that the net converges to u. 4 4. 5. a) Prove that a topological space ( X , ) is T2 if and only if every net in X converges to at most one point in X. 3 b) State and prove Tietze Extension theorem. 7 a) Define compactness. Is compactness a hereditary property ? Answer with reasons. Prove that the product space (X Y , ) is connected if and only if ( X , ) and (Y , ) are connected. 6 Define a net. Prove that in a topological space ( X , ) , a point u X is a limit point 7. Prove that continuous image of a compact space is compact. 3 c) Define the one-point compactification ( X u , u ) of a non-compact locally compact Prove that every component of a topological space is closed. 1 a) Using connectedness, prove that any continuous function f : [ 0,1] [ 0,1] has a fixed point u i.e. f ( u ) u . b) a) In a topological space ( X, ) if A {G X that is a topology on X. is [ P.T.O. : for each x G , there is a member U v such that U ( x ) G } . Prove connected and A B A then prove that B is connected. 4 PG-Sc.-1310-G 5 Define a uniformity on a non-void set X. Let a uniform space. Define ( X ,v ) be T2 space ( X , ) . Then show that ( X u , u ) is a compact T2 space. 5 6. 5 c) 2 b) 4 PGMT-6A (PT/10/VIA) PG-Sc.-1310-G 5