Important Definitions
1 . Point Sets Point sets are sets whose elements are points or vectors in E n or R n ( n-dimensional euclidean space ).
2 . Hypersphere A hypersphere in E n with centre at ‘ a ’ and radius ∈ > 0 is defined to be the set of points
X = - { x :| x — a | = ∈ }
3 . An ∈ neighbourhood An & neighbourhood about the point ‘ a is defined as the set of points lying inside the hypersphere with centre at ‘ a ’ and radius ∈ > 0 .
4 . An Interior Point A point ‘ a ’ is an interior point of the set S , if there exists an ∈ neighbourhood about ‘ a ’ which contains only points of the set S .
5 . Boundary Point A point ‘ a ’ is a boundary point of the set S if every ∈ neighbourhood about ‘ a ’ contains points which are in the set and the points which are not in the set .
6 . An Open . Set A set S is said to be an open set , if it contain only the interior points .
7 . A Closed Set A set S is said to be a closed set , if it contains a its boundary points .
8 . Lines In E n the line through the two points x1 and x2 , x1 ≠ x2 is defined to be the set of points . X = { x : x = λ x1 + ( 1 — λ ) x2 , for all real λ }
9 . Line Segments In En , the line segment joining two point x1 and x2 is defined to be the set of points .
X = { x : x = λ x1 + ( 1 — λ ) x2 , 0 ≤ λ ≤ 1 }
10 . Hyperplane A hyperplane is defined as the set of points satisfying
c1x1 + c2x2 + …+ cnxn = z ( not all ci = 0 ) or cx = z for prescribed values of c1 , c2 ,…, cn and z .