Download Chapter 13 Topological Appendix

Chapter 13
Topological Appendix
A typical reader of this textbook is supposed to have some basic knowledge of topology and functional analysis. Nevertheless, for the reader’s convenience, in this chapter we enumerate some
useful facts and assertions from these fields.
We introduce also some less known (but basic) notions - e.g. a notion of a semi-open set - that
usually are not mentioned in basic courses. On the other hand some notions are mentioned without
being defined, some of them may be defined in a more precise way in specialized chapters.
13.1 Topological space, open and closed sets etc.
A topological space is a pair (X, T ) of a set X and a collection T of subsets of X, called open sets,
satisfying these properties:
(i) X and the empty set are open sets.
(ii) The finite intersection of open sets is an open set.
(iii) The union of any number of open sets is an open set.
In a topological space we define a set to be closed if its complement is an open set. For each set A
in a topological space we define cl A (closure of A) as an intersection of all closed sets Z such that
A ⊂ Z. The set cl A is closed and is called the closure of A. We define a boundary of A (Bd A) by
Bd A = cl A ∩ cl (X − A).
Separation Axioms 13.1. A topological space (X, T ) is
T1
if for all x ∈ X
T2 or Hausdorff
T3
cl {x} = {x}
if ∀a, b ∈ X, a , b ∃ O1 , O2 ∈ T, a ∈ O1 , b ∈ O2 and O1 ∩ O2 = ∅
if a ∈ X, B ⊂ X is closed, a < B, there exist O1 , O2 ∈ T, a ∈ O1 , B ⊂ O2 and O1 ∩ O2 = ∅
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13.1 Topological space, open and closed sets etc.
T4
146
if A, B ⊂ X are disjoint and closed, there exist O1 , O2 ∈ T, A ⊂ O1 , B ⊂ O2 and
O1 ∩ O2 = ∅
If X is a topological space and p is a system of open subsets of X, we say that p is an open
covering of X (or an open cover of X) if
[
U=X
U∈p
is true.
Definition 13.2. A Hausdorff space X is said to be paracompact if every open covering of X admits
a locally finite open refinement.
Definition 13.3.
(a) A family {eα }α∈A of nonnegative continuous functions on a topological space X is said to be
P
a locally finite partition of unity if for all x ∈ X α∈A eα (x) = 1, and for every x ∈ X, there
exist a neighbourhood W(x) and a finite subset A(x) ⊂ A such that for all y ∈ W(x) and for
all α ∈ A \ A(x), eα (y) = 0.
(b) A locally finite partition of unity {eα }α∈A is said to be inscribed into an open covering {Gγ }γ∈Γ
of a topological space X if for any α ∈ A, there exists γ ∈ Γ such that
supp(eα ) = Cl{x ∈ X | eα (x) > 0} ⊂ Gγ .
It is easy to see that for a locally finite partition of unity {eα }α∈A inscribed into a covering
{Gγ }γ∈Γ , the family of open sets {e−1
α ((0, 1])}α∈A gives a locally finite refinement of the covering
{Gγ }γ∈Γ .
The following proposition is proved for example in [45]:
Proposition 13.4. A Hausdorff space X is paracompact if and only if each open covering of X
admits a locally finite partition of unity inscribed into this covering.
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13.2 Nets in topological spaces
147
13.2 Nets in topological spaces
We say a set Γ is directed by a relation ≥ if the relation is reflexive, transitive, and for all γ1 , γ2 ∈ Γ
there exists γ3 ∈ Γ such that γ1 ≤ γ3 and γ2 ≤ γ3 is true. Both the positive integers and the set
of real numbers are directed by the usual order. The system of all neighbourhood of a point in a
topological space is directed by reverse inclusion.
A net in a set X based on a directed set Γ is a function x : Γ → X. In particular, a sequence is a
net, where Γ = Z+ with the usual order. In what follows we will write {xγ }γ∈Γ for x : Γ → X, and
xγ for x(γ).
Let {xγ }γ∈Γ be a net in a topological space X. We say that l ∈ X is a limit point of {xγ }γ∈Γ if for each
neighbourhood O of l there exists an index γ0 ∈ Γ such that for all γ ≥ γ0 we have xγ ∈ O. In a
Hausdorff space a net can have at most one limit point. If l is a limit point of a net, we say that this
net converges to l. We write
lim xγ = l.
γ∈Γ
The following notion of the strong convergence of functions was defined by Kupka and Toma:
Definition 13.5. Let X be an arbitrary set, let (Y, T ) be a topological space. Be { fγ }γ∈Γ a net of
functions from X to Y. We say that { fγ }γ∈Γ converges to a function f : X → Y strongly if for every
open cover C of Y there exists an index α ∈ Γ such that for every x ∈ X and every γ ≥ α there
exists an open set U ∈ C such that { f (x), fγ (x)} ⊆ U.
Remark 13.6. It is known that if X is a topological space and Y a regular topological space and if
{ fγ }γ∈Γ is a net of continuous functions from X to Y converging strongly to a function f : X → Y,
then f is continuous. Later we will provide a generalization of this result for multifunctions.
13.3 Semi-open sets and quasicontinuity
In what follows, X and Y will be topological spaces.
A subset B of X is said to be semi-open if there exists an open set G such that G ⊂ B ⊂ cl G or,
which is equivalent, B ⊂ cl (int B) ([31]).
We will use the following properties of semi-open sets ([31], [41]):
(1) The union of any family of semi-open sets is semi-open.
(2) The intersection of an open set and a semi-open one is semi-open.
(3) A set S is semi-open if and only if S = (int S ) ∪ B where B is a subset of the boundary of
int S .
Of course, every open set is also semi-open. Every interval in R is a semi-open set.
Levine in [31] defined a function f : X → Y to be semi-continuous if f − (G) is semi-open for
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13.4 Property of Baire, Baire space
148
any open subset G of Y. Kempisty in [21] introduced the notion of quasicontinuous function. A
function f : X → Y is said to be quasicontinuous at x ∈ X if, for any open set V such that f (x) ∈ V
and any open U such that x ∈ U, there exists a non-empty open set W ⊂ U such that f (W) ⊂ V. If
f is quasicontinuous at each point x ∈ X it is said to be quasicontinuous in X. A. Neubrunnová in
[40] proved that a function f : X → Y is semi-continuous if and only if it is quasicontinuous.
We say that a set S is a semi-neighbourhood of a point x if there exists a semi-open set A such
that x ∈ A ⊂ S holds. It is easy to see (using [21] and [31]) that a function f : X → Y is
quasicontinuous if and only if for each x in X and for each open G ⊂ Y such that f (x) ∈ G the set
f − (G) is a semi-neighbourhood of x.
13.4 Property of Baire, Baire space
In what follows we define the notions of a nowhere dense set and of a set of first category. These
sets are - in a way - "topologically small".
Definition 13.7. Let (X, T ) be a topological space. We say that a set A ⊂ X is nowhere dense if
int A = ∅ is true.
We say that a set B ⊂ X is of first category (or meagre) if it can be expressed as the union of
countably many nowhere dense subsets of X.
Remark 13.8. Every finite subset of Rn is nowhere dense, every countable subset of Rn is of the
first category.
Definition 13.9. Be (X, T ) a topological space. A set B ⊂ X is said to have the property of
Baire (Baire property) if there exists an open set U ∈ T and two meagre sets S 1 , S 2 such that
B = (U − S 1 ) ∪ S 2 .
Definition 13.10. A topological space (X, T ) is called a Baire space if every meagre subset A ⊂ X
has an empty interior.
Remark 13.11. The property defined above is equivalent with the following one:
Every nonempty open set is of second category in X.
13.5 Topological spaces, metric spaces, normed spaces
We suppose that the notions of distance (metric), metric space and Cauchy sequence in a metric
space are known to the reader. A subset S of a metric space is called a complete set if every Cauchy
sequence in S converges to a point from S .
Definition 13.12. Let X be a set, (Y, d) a metric space, { fγ }γ∈Γ a net of functions from X to Y and
let f : X → Y be a function. We say that { fγ }γ∈Γ converges to f uniformly if
∀ > 0∃γ0 ∈ Γ, ∀γ ≥ γ0 , ∀x ∈ X : d( f (x), fγ (x)) < .
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13.5 Topological spaces, metric spaces, normed spaces
149
It was proved that if Y is a metric space, the strong convergence of a net of functions { fγ }γ∈Γ
from X to Y to a function f implies the uniform convergence of this net to f .
The following theorem will be later generalized for the case of multifunctions.
THEOREM 13.13. Be (X, T ) a topological space, (Y, d) a metric space. Be x0 ∈ X and { fγ }γ∈Γ be
a net of functions from X to Y that are all continuous at x0 . Let { fγ }γ∈Γ converges uniformly to a
function f : X → Y. Then f is continuous at x0 .
Proof. We will prove
∀ > 0∃O ∈ T : x0 ∈ O and ∀x ∈ O d( f (x), f (x0 )) < .
(?)
Choose an > 0. We know that there exists γ0 ∈ Γ such that
∀γ ≥ γ0 ∀x ∈ X : d( fγ (x), f (x)) < .
3
Pick an α > γ0 . Since fα is continuous at x0 we have
∃O ∈ T : x0 ∈ O and ∀x ∈ O : d( fα (x), fα (x0 )) < .
3
The triangle inequality, (I) and (II) imply together:
∀x ∈ O d( f (x), f (x0 )) ≤ d( f (x), fα (x0 )) + d( fα (x), fα (x0 )) + d( fα (x0 ), f (x0 )) <
+ + = .
3 3 3
Thus f is proved to be continuous at x0 .
(I)
(II)
In what follows, if X and Y are topological spaces, by C(X, Y) we denote the set of all continuous functions from X to Y. If Y is a metric space with a metric d, by C(X, (Y, d)) we mean the set
of all continuous functions from X to (Y, d), equipped with the (infinite valued) metrics ρ, where
ρ( f, g) := sup d( f (x), g(x))
x∈X
is the distance of two points f and g from C(X, Y). It is well known that the convergence in
C(X, (Y, d)) coincides with the uniform convergence in C(X, Y).
Now we are going to define the notion of the Hausdorff distance.
Definition 13.14. If (X, d) is a metric space, > 0, A ⊂ X, we denote
B (A) = {x ∈ X : ∃a ∈ A d(x, a) < }.
Let C and K be two subsets of X. The number
H(C, K) = inf{ > 0 : B (K) ⊃ K and B (K) ⊃ C}
is called the Hausdorff distance between C and K.
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13.5 Topological spaces, metric spaces, normed spaces
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13.5.1 Normed spaces, Banach spaces
A topological vector space E is called a normed space if its topology is generated by some norm,
i.e. by some mapping p : E → [0, ∞) with properties:
(i) (p(x) = 0) ⇔ (x = 0);
(ii) p(x + y) ≤ p(x) + p(y); and
(iii) p(λx) = |λ|p(x);
for all scalars λ and x, y ∈ E. We denote by ||.|| = p and ||x|| the norm p(x) of an element x ∈ X. A
normed space (E, ||.||) is called a Banach space if it is complete under the metric ρ(x, y) = ||x − y||,
induced by the norm ||.||.
In our textbook we work with some of these Banach spaces (let A be a set):
(i) l∞ [A] = {x : A → R | x is bounded and ||x||∞ = sup{|x(α)| | α ∈ A}}
(ii) l p [A] = {x : A → R | {α ∈ A | x(α) , 0} is at most countable and ||x|| p = (
∞}, p ≥ 1
P
α
|x(α)| p )1/p <
(iii) C0 [A] = {x : A → R | x ∈ l∞ [A] and for every > 0 the set {α ∈ A | |x(α)| ≥ } is finite}.
Exercises
Exercise 1 Define a topological space (X, τ) and a net {xγ }γ∈Γ in X such that {xγ }γ∈Γ has (a) exactly
two limit points, (b) infinitely many limit points.
Exercise 2 Consider R with its natural topology. Decide which of the following sets are semiopen.
(a) [1, 2)
(b) ∅
(c) ∪∞
i=0 Oi , where O0 = {0}, and
!
1 1 1 1
− , +
.
Oi =
i i2 i i2
(d) R \ S , where S is any nowhere dense subset of R.
Exercise 3 Let (X, τ) be a topological space, A ⊂ X be nowhere dense. Denote B = X \ A. Show,
that for every semiopen set S in X the set S ∩ B is semiopen.
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13.5 Topological spaces, metric spaces, normed spaces
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Exercise 4 Let (X, τ) be a topological space. Prove that any closed subset A ⊂ X can be represented
as A = B ∪ C, where B is a semiopen set and C is a nowhere dense set.
Exercise 5 Find two quasicontinuous functions f : R → R, g : R → R such that f + g will not
be quasicontinuous.
Exercise 6 Let X be a metric space,
cl(X) = {A ⊂ X; A is closed, A , ∅}.
Prove that the function
H : cl(X) × cl(X) → R,
where H(C, K) is the Hausdorff distance between C and K for C, K ∈ cl(X), is a metric on cl(X).
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