www.pdfquestion.in

Name of the University : Vardhman Mahaveer Open University
Degree : M.Sc
Department : Mathematics
Subject Code/Name : MAMT-02 – Real Analysis and Topology
Year : I
Document Type : Question Bank
Website : vmou.ac.in

Download Model/Sample Question Paper :

Question 1 : https://www.pdfquestion.in/uploads/vmou.ac.in/3600.-MAMT-02_45.pdf
Question 2: https://www.pdfquestion.in/uploads/vmou.ac.in/3600.-MAMT-02_46.pdf

Real Analysis & Topology :

Section – A :
1. Define a denumerable set.
A. A set S is said to be a denumerable set if it is equipotent to the set N

Related : VMOU MAMT01 Advanced Algebra M.Sc Question Bank : www.pdfquestion.in/3596.html

---

More Related Posts:

• MAMT01 Advanced Algebra M.Sc Question Bank : vmou.ac.in
• MAMT06 Analysis and Advanced Calculus M.Sc Question Bank : vmou.ac.in
• MAMT05 Mechanics M.Sc Question Bank : vmou.ac.in
• MAMT09 Integral Transforms and Integral Equations M.Sc Question Bank :…

2. Define an algebra of sets.
3. The outer measure of the set of natural number is :
4. Define measurable sets.
6. What is measurable function-
7. What is step function-
8. Define uniform convergence.
9. What is Bernstein polynomial-
10. What is Lebesgue integral function.
11. Define square summable function.
12. Define Cauchy sequence in –
13. Define scalar product in – .
14. What is closed orthonormal system-
15. Define convergence in norm in – space.
16. State Minkowski’s inequality.
17. Define co-countable topology.
18. What is comparable topologies-
19. What is closed set-
20. Define neighbourhood.
21. Define Hereditary property.
22. Define base for a topology.
23. What is first countable-
24. What is an open mapping-
25. Define topological property.
26. Define second countable space.
27. Define – space.
28. What is a regular space-
29. What is an open cover of a set-
30. Define a compact space.
31. Define locally compact space.
32. What is Bolazano weirstrass property-
33. What is finite intersection property-
34. The one point compactification of the interval (0, 1) is :
35. The one point compactification of the set complex numbers is called :
36. What is separated sets-
37. What is a disconnected set –
38. Define locally connected space at a point.
39. What is a disconnection-
40. Define a component-
41. Define product topology.
42. What is the projection mappings-
43. Define finitely short family.
44. What is a Quotient space-
45. What is Embedding-
46. Define a residual set.
47. Define eventually net.
48. What is Ultranet-
49. What is filter-
50. Define subbase of a filter.

Section B :
1. Describe cantor set.
2. Prove that every open interval is a Barel set.
3. Show that outer measure is translation invariant.
4. The lower Lebesgue Darboux sums of any bounded measurable function f on a measurable set E can not exceed its upper Lebesgue Darboux sums.
5. Show that every bounded measurable functions f defined on a measurable set E is L-integrable on E.
6. Show that homeomorphism is an equivalence relation in the family of topological spaces.
7. The property of a space being a Hausdroff space is a hereditary property.
8. Show that regularity is a topological property.
9. Show that a closed sub space of normal space is a normal space.
10. A closed subset of a compact space is compact.
11. Show that a compact space has Bolzano-Weiers trass property.
12. A compact Hausdorff space if normal.
13. Show that every compact topological space is locally compact, but converse is not necessarily true.
14. Every open continuous image of a locally compact space is locally compact.
15. The one point compactification of the plane is homeomorphic to the sphere.
16. Two closed subsets of a topological space are separated iff they are disjoint.
17. A topological space X is disconnected iff A is the union of two non-void disjoint open (closed) sets.
18. Give an example of a locally connected space which is not connected.
19. The image of a locally connected space under a open continuous mapping is locally connected.
20. Show that every filter J on a non-void set X is contained in an ultrafilter on X.
21. Show that every subnet of an ultranet if an ultranet.

Analysis and Advanced Calculus :
1. Define Norm and write the set of axions of Normal linear space.
2. Write the Summability for a series S –
3. If N be a Normed linear space and x, y, – – then Prove
4. Show that every normed linear space is a metric space.
5. If N be a normed linear space with the norm – . – , then prove that mapping
6. Show that every convergent sequence in a normal linear space.
7. Show that the limit of a convergent sequence is unique.
8. Write the Reflexive, Symmetric and Transitive relations for factor –
9. Show that the linear spaces R(real) and (Complex) are normed linear spaces
10. If T be a linear transformation of a normed linear space N –