Page 83 - DMTH401_REAL ANALYSIS

Unit 5: Connectedness

 C connected so cannot be separated by U,V, so does not meet one of them. Notes
 C component of U, x ÎC. Find  > 0 with B(x, )  U. B(x, ) connected and C union of
all connected subsets of U containing x so B(x, )  C, so C open.

5.8 Keywords

Path Connected: T path connected if any two points can be joined by a path.
Topological Invariant: Number of connected components is topological invariant.

5.9 Review Questions

1. Define Connectedness.
2. Discuss the Connectedness in metric spaces.

3. Explain connected spaces from others.
4. Describe connected components and Path connected.

Answers: Self Assessment

1. T = AB 2. disjoint open
3. I  4. T connected

5. K connected 6. Equivalence classes

5.10 Further Readings

Books Walter Rudin: Principles of Mathematical Analysis (3rd edition), Ch. 2, Ch. 3.
(3.1-3.12), Ch. 6 (6.1 - 6.22), Ch.7(7.1 - 7.27), Ch. 8 (8.1- 8.5, 8.17 - 8.22).
G.F. Simmons: Introduction to Topology and Modern Analysis, Ch. 2(9-13),
Appendix 1, p. 337-338.
Shanti Narayan: A Course of Mathematical Analysis, 4.81-4.86, 9.1-9.9, Ch.10,Ch.14,
Ch.15(15.2, 15.3, 15.4)
T.M. Apostol: Mathematical Analysis, (2nd Edition) 7.30 and 7.31.
S.C. Malik: Mathematical Analysis.
H.L. Royden: Real Analysis, Ch. 3, 4.

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