Page 32 - DMTH503_TOPOLOGY

Topology

Notes  Let (X, T) be a topological space. A  X is called a neighbourhood of a point x  X if
 G  T with x  G such that G  A.

 Let (X, T) be a topological space. A  X is said to be dense or everywhere dense in X if
A = X.
 If A said to be nowhere dense set in X if int ( A ) = .
 Let {X, T} be a topological space and A  X then X is said to be separable if
(i) A = X (ii) A is countable

 Let (X, T) be a topological space and A  X.
A point x  X is said to be the limit point if each open set containing x contains at least one
point of A different form x.

 The set of all limit point of A is called derived set of A.
 Let (X, T) be a topological space and A  X. A point x  A is called a interior point of A iff
there exists an open set G such that x  G  A. It is denoted by Int (A) or A°.
C
 A point x  X is called an exterior point of A iff it is an interior point of A or X – A. It is
denoted by ext (A).

1.9 Keywords

Complement: The complement of a set A w.r.t. the universal set X is defined as the set X-A and is
denoted by A°.

symbolically, A° = X – A = {x  X : x  A}.
Intersection: The intersection of two sets A and B, denoted by A  B, is
A B = {x : x  A and x  B}
Subset: If every element of set A is also an element of set B, then A is called a subset of B. It is
denoted by the symbol A B.
Superset: A B is also expressed by writing, B A.
Union: The union of two sets A and B, denoted by A B, is
A B = {x : x A or x  B}

1.10 Review Questions

1. Let X = {a, b, c, d, e, f}, which of the following collections of subsets of X is a topology on
X? (Justify your answers).
(a) T = {X, , {c}, {b, d, e}, {b, c, d, e}, {b}};
1
(b) T = {X, , {a}, {b, d, e}, {a, b, d}, {a, b, d, e}};
2
(c) T = {X, , {b}, {a, b, c}, {d, c, f}, {b, d, e, f}}.
3
2. If X = {a, b, c, d, e, f} and T is the discrete topology on X, which of the following statements
are true?
(a) X  T (b) {X}  T
(c) {}  T (d)   T
(e) {}  X (f) a  T

26 LOVELY PROFESSIONAL UNIVERSITY