Alagappa University Topology II M.SC Model Question Papers : alagappauniversity.ac.in

Name of the University : Alagappa University
Degree : M.SC
Department : Mathematics
Subject Code/Name : Topology II
Year : II
Semester : IV
Document Type : Model Question Papers
Website : alagappauniversity.ac.in

Download Model/Sample Question Paper : APRIL 2010 : https://www.pdfquestion.in/uploads/alagappauniversity.ac.in/3892-MSC_MATHS.pdf

Alagappa University Topology II Question

Duration : 3 Hours
Maximum : 75 Marks
Part – A (10 × 2 = 20) :

Related : Alagappa University Algebra-I M.Sc Model Question Papers : www.pdfquestion.in/3996.html

April 2011

Part – A : (10 × 2 = 20)
Answer all the Questions
1. Define a locally compact space. Give an example.
2. Define one point compactification of a locally compact Hausdorff Space.
3. Define a completely regular space. Give an example.
4. When do you say two compactifications of a compact Hausdorff space are equivalent ? Give an example.
5. In a metric space prove that every closed set is a G – set.

6. Define Open refinement of a collection of subsets of a space X. give an example.
7. Define a complete metric space and give an example.
8. Define a totally bounded metric space and give an example.
9. Is the collection {fn}, where fn(x) = x + sin x, of subset of C(R, R) pointwise bounded ? Justify.
10. Is a open subset of a Baire space a Baire space ? Justify.

Part – B : (5 × 5 = 25)
Answer all the Questions
11 a. Let X be a Hansdorff space. Prove that X is locally compact at x if, and only if, for every neighbourhood U of x, there is a neighbourhood V of x such that V is compact.
12 a. Show that every locally compact Hausdorff space is completely regular.
13 a. Let X be a metrizable space. Prove that X has a basis that is countably locally finite.
14 a. Let (X, d) be a metric space. Prove that there is an isometric embedding of X into a complete metric space. (Or)
14 b. Let (X, d) be a metric space. If X is complete and totally bounded, prove that X is compact.
15 a. If X is a compact Hausdorff space or a complete metric space prove that X is a Baire space.
17. Define Stone-Cech compactification of a space X. Establish (i) the extension property and (ii) uniqueness of Stone-Cech compactification.
18. Let X be a regular space with a basis that is countably locally finite. Prove that X is metrizable.
19. State and prove Peano space – filling curve theorem.
20. State and prove Ascoli’s theorem.

April 2010

M.Sc. Degree Examination,  :
Analysis – I :
(CBCS—2008 Onwards)
Duration : 3 Hours
Maximum : 75 Marks
Section – A (10 × 2 = 20)
Answer All questions.
1. Define a metric space and give an example.
2. Define a connected set and give an example.
3. Define a complete metric space and give an example.
4. If the sequence {pn} in a metric space X converges prove that it is bounded.
5. If an converges absolutely prove that an converges.
6. Define a power series and give an example.
7. Define a continuous function and give an example.
8. Define discontinuity of second kind and give an example.
9. If f is differentiable at a point prove that it is continuous at that point.
10. Define local minimum.

Section – B : (5 × 5 = 25)
Answer All questions.
11 a. Prove that a set E is open if, and only if, its complement is closed.
12 a. If X is a metric space prove that every convergent sequence is a Cauchy sequence. Further, if X is a compact metric space and if {pn} is a Cauchy sequence in X prove that {pn} converges to some point of X.
13 a. State and prove Merten’s theorem on the product of two non absolutely convergent series.
14 a. Prove that a mapping f of a metric space X into a metric space Y is continuous if, and only if, f-1(C) is closed in X for every closed set C in Y. (Or)
b. If f is a continuous mapping of a metric space X into a metric space Y. If E is a connected subset of X prove that f(E) is connected.
15 a. State and prove intermediate value theorem. (Or)
b. State and prove the Chain rule for differentiation.

Section – C : (3 × 10 = 30)
Answer any Three questions
16. Define a k-Cell. Prove that every k-Cell is compact.
17. State and prove the root test.
18. State and prove Riemann’s theorem on rearrangement of series.
19. a) Let f be a continuous mapping of a metric space X into a metric space Y. Prove that f is uniformly continuous on X.
b) Prove that the continuous image of a compact space is compact.
20. State and prove Taylor’s theorem.