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.