Birla Institute of Technology and Sciences, Pilani MATH F 311 Introduction To Topology Problem Set-2 (Article 13-17)
Birla Institute of Technology and Sciences, Pilani MATH F 311 Introduction To Topology Problem Set-2 (Article 13-17)
Birla Institute of Technology and Sciences, Pilani MATH F 311 Introduction To Topology Problem Set-2 (Article 13-17)
Q.4 Let (X, T) be a topological space and let A, B be non-empty subsets of X Then show that:
(a) ' (d) ( A B) ' A' B '
(b) x A' x ( A {x}) ' (e) ( A B) ' A' B '
(c) A B A' B ' (f) A B A B
(g) ( A ) A (h) ( A B) A B
(i) ( A B) A B
where A’ means derived set of A and A means interior of A.
Q.5 Show that every derived set in a topological space is a closed subset of X.
Q.7 Let X = { p, q, r, s, t} and T = { Φ, X, {p}, {p, q}, {p, q, t}, {p, r, s}, {p, q, r, s}} be the topology on
X. Determine closure, interior and derived set of following sets:
(a) A = {r, s, t} (b) B = {q} (c) C = {q, s, t}
Q.8 Let (X, T) be a topological space and Y X . Show that if A Y then T-interior of A is subset of
interior of A in induced subspace topology on Y.
Q.9 Show that all intervals (a, 1] and [0, b) where 0 < a, b < 1, form a sub-base for the relative usual
topology induced on closed interval I = [0, 1]
Q.10 Let Y be a subspace of a topological space X and Z be a subspace of Y then show that Z is also a
subspace of X.