Page 34 - DMTH503_TOPOLOGY

Topology

Notes 20. Let (X, T) be a topological space and A  X. A point x of A is an interior point of A iff it is
not a limit point of X – A.
21. Let T = {X, , {p}, {p, q}, {p, q, t}, {p, q, r, s}, {p, r, s}} be the topology on X = {p, q, r, s, t}.
Determine limit points, closure, interior, exterior and boundary of the following sets:
(a) A = {r, s, t} (b) B = {p}
22. It T = {, {a}, {a, b}, {a, c, d}, {a, b, e}, {a, b, c, d}, X} be a topology on X = {a, b, c, d, e} then

(a) Point out T-open subsets of X.
(b) Point out T-closed subsets of X.
(c) Find the closure of the sets {a}, {b}, {c}.
(d) Find the interior points of the subset A = {a, b, c} on X.
(e) Which of the sets {a}, {b}, {c, e} are dense in X?

Answers: Self Assessment

1. T = {, X}, T = {, X, {b}}, {a, b}, T = {, X, {a}, {b}, {a, b}}.
1 2 3
2. Yes
8. nhd of r are {p, r, s}, {p, q, r, s}
nhd of t is {p, q, t}

12. D(A) = {c}
13. D ({b}) = D ({a, b}) = D({b, c}) = D({c, a}) = {c}

1.11 Further Readings

Books J. L. Kelley, General Topology, Van Nostrand, Reinhold Co., New York.
S. Willard, General Topology, Addison–Wesley, Mass. 1970.

28 LOVELY PROFESSIONAL UNIVERSITY