Nonlinear Control Sy.. - Free

44 CHAPTER 2. MATHEMATICAL PRELIMINARIES
2.5 Basic Topology
A few elements of basic topology will be needed throughout the book. Let X be a metric
space. We say that
(a) A neighborhood of a point p E X is a set Nr (p) C X consisting of all points q E X
such that d(p,q) < r.
(b) Let A C X and consider a point p E X. Then p is said to be a limit point of A if
every neighborhood of p contains a point q # p such that q E A.
It is important to notice that p itself needs not be in the set A.
(c) A point p is an interior point of a set A c X if there exist a neighborhood N of p
such that N C E.
(d) A set A c X is said to be open if every point of A is an interior point.
(e) The complement of A C X is the set A` = {p E X : p V A}.
(f) A set A C X is said to be closed if it contains all of its limit points. Equivalently, A
is closed if and only if and only if A` is open.
(g) A set A C X is bounded if there exist a real number b and a point q E A such that
d(p, q) < b, dp E A.
2.5.1 Basic Topology in 1R"
All the previous concepts can be specialized to the Euclidean space 1R". We emphasize those
concepts that we will use more frequently. Now consider a set A C 1R".
Neighborhood: A neighborhood of a point p E A C R" is the set Br (p) defined as
follows:
Br(p) = {x E R' :Iix - p1I < r}.
Neighborhoods of this form will be used very frequently and will sometimes be referred
to as an open ball with center p and radius r.
Open set: A set A C 1R" is said to be open if for every p E A one can find a
neighborhood Br(p) C A.
Bounded set: A set A C ]R" is said to be bounded if there exists a real number
M > 0 such that
IIxli < M Vx E A.