Topological properties for sets and functions

Chapter 2 Topological properties for sets and functions
Chapter 2.
Topological properties for sets and functions
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Chapter 2 Topological properties for sets and functions
Relative Interior of a Convex Set
The interior of a subset C of IR
n
is the union of all open sets (of IR
n
)
contained in C. Since any union of open sets is open, the interior is also
the largest open set (of IR
n
) contained in C. The interior of C is denoted
by intC . From this definition, we have
x ∈ int C ⇔ ∃δ > 0 such that B(x, δ) ⊆ C .
Many nonempty convex sets have an empty interior. For example, in IR
2
,
int [a, b] = ∅. Similarly, in IR
3
, the interior of a triangle ABC is empty. So
our aim is to define a substitute to the interior of a convex set, called the
relative interior, in such a way that the relative interior of any nonempty
convex set is nonempty.
To define the relative interior, we need the concept of affine set.
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Chapter 2 Topological properties for sets and functions
Affine Sets. Definition
Let A be a nonempty subset of IR

n
. A is affine if
∀x, y ∈ A, ∀α ∈ IR αx + (1 − α)y ∈ A.
The vector αx + (1 − α)y is called an affine combination of x and y.
The line passing through two points x and y is defined by
{αx + (1 − α)y | α ∈ IR}.
1) Geometrically, A is affine if it contains the line passing through each
pair of its points.
2) A singleton, a line, a plane are affine sets
3) an affine set is convex.
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Chapter 2 Topological properties for sets and functions
Affine Combination
An affine combination of finitely many points x
1
, . . . , x
k
is defined by
k

i=1
α
i
x
i
where
k

i=1
α

i
= 1.
As for convex sets, we have the following characterization
Proposition. A subset A of IR
n
is affine if and only if it contains
every affine combination of finitely many of its points.
It is immediate that the translation of an affine set A, namely A + x with
x ∈ IR
n
, is affine. More specifically, the affine sets are just translates of
subspaces.
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Chapter 2 Topological properties for sets and functions
Linear Subspaces
Let us recall that a subspace L is a subset of IR
n
which satisfies the
property :
∀x, y ∈ L, ∀α ∈ IR, x + y ∈ L and αx ∈ L
and that two affine sets A and B are parallel if there exists x ∈ IR
n
such
that A = B + x.
Proposition. The following statements hold :
(i) L is a subspace if and only if L is affine and 0 ∈ L.
(ii) Let A be an affine set. Then there exists a unique subspace L parallel
to A. Moreover one has A = L + a for every a ∈ A.
(iii) The translate of a subspace is an affine set.
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Chapter 2 Topological properties for sets and functions
Dimensions and Hyperplanes
The dimension of an affine set A is the dimension of its parallel
subspace. An affine set of dimension 0 is called a point, an affine set of
dimension 1, a line, an affine set of dimension 2, a plane and an affine
set of dimension n − 1, an hyperplane.
An hyperplane is defined by the set
H = {x ∈ IR
n
| x
∗
, x = b}
where x
∗
∈ IR
n
, x
∗
= 0, and b ∈ IR.
x
∗
is called the normal vector to H.
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Chapter 2 Topological properties for sets and functions
Affine Hull
Let C be a subset of IR
n
. The affine hull of C, denoted aff C , is the
intersection of all affine subsets of IR
n

containing C. The affine hull is
affine.
Proposition. The affine hull of C is the set of all affine combinations
of finitely many points of C.
Let C = {x
1
, x
2
, . . . , x
k
}. Then
aff C =

k

i=1
α
i
x
i




k

i=1
α
i
= 1


Examples : aff {x} = {x}, aff [x, y ] is the line generated by x and y.
aff B(0, 1) = IR
n
.
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Chapter 2 Topological properties for sets and functions
Affinely Independent Points
Definition. Let S = {x
0
, . . . , x
k
} be a set of k + 1 points of IR
n
. The
points of S are affinely independent if aff S has dimension k
Proposition. Let S = {x
0
, . . . , x
k
} be a set of k + 1 points of IR
n
.
The points of S are linearly independent if and only if the vectors
x
1
− x
0
, x
2

− x
0
, . . . , x
k
− x
0
are linearly independent.
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