Download Introduction to Functions, Sequences, Metric and Topological

Introduction to Functions, Sequences, Metric
and Topological Spaces, Continuity,
Semicontinuity, and Hemicontinuity
Tigran A. Melkonyan, University of Nevada, Reno
August 6, 2007
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Functions
De…nition 1 A function f from (or on) a set X to (or into) a set Y is a
rule that assigns to each x 2 X a unique element f (x) in Y:
De…nition 2 The collection G of pairs (x; f (x)) in X Y is called the graph
of the function f:
The word “mapping”is often used for “function”.
We will express the fact that f is a function from X into Y by writing
f : X ! Y:
De…nition 3 The set X is called the domain of f: The set of values taken
by f; that is, the set
fy 2 Y : 9x such that y = f (x)g
is called the range of f:
De…nition 4 For A X; the image f (A) of A under f is the set of elements of Y such that y = f (x) for some x 2 A :
f (A) = fy 2 Y : 9x 2 A such that y = f (x)g:
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De…nition 5 Function f is onto Y if and only if Y = f (X):
De…nition 6 For B
Y; the inverse image f
those elements x 2 X for which f (x) 2 B :
f
1
1
(B) of B is the set of
(B) = fx 2 X : f (x) 2 Bg:
De…nition 7 Function f is called one-to-one Y (or univalent, or injective)
if and only if f (x1 ) = f (x2 ) only if x1 = x2 :
De…nition 8 Functions f : X ! Y which are one-to-one and onto are
called one-to-one correspondences (or bijective).
De…nition 9 If function f is a one-to-one correspondence between X and Y
then there exists function g : Y ! X such that g(f (x)) = x and f (g(y)) = y:
The function g is called the inverse of f and is frequently denoted by f 1 :
De…nition 10 Let f : X ! Y and g : Y ! Z: The function h : X ! Z
de…ned as h(x) = g(f (x)) for all x 2 X is called the composition of g with
f and is denoted by g f:
De…nition 11 Let f : X ! Y and A X: The function g : A ! Y de…ned
as g(x) = f (x) for all x 2 A is called the restriction of function f to A
and is sometimes denoted by f jA:
Exercise Set 1
Exercise 1: Let f : X ! Y: Show that f is onto if there is a mapping
g : Y ! X such that f g is the identity map on Y; that is, f (g(y) = y for
all y 2 Y:
Exercise 2: Show that:
S
S
(i) f
A =
f [A ] ;
2
(ii) f
T
2
2
A
T
f [A ] ;
2
2
T
(iii) Provide an example where f
A
2
Exercise 3: Show that:
S
S 1
f [A ] ;
(i) f 1
A =
2
(ii) f
1
T
2
2
6=
T
f [A ]
2
2
A
=
T
f
1
[A ] ;
2
Metric Spaces
De…nition 12 A metric on a nonempty set X is a function d : X X ! <+
satisfying the following properties:
a) Non-negativity: For all x; y 2 X; d(x; y) 0:
b) Discrimination:
d(x; x) = 0 and
d(x; y) = 0 ! x = y:
c) Symmetry: For all x; y 2 X; d(x; y) = d(y; x):
d) Triangle Inequality: For all x; y; z 2 X; d(x; z)
d(x; y) + d(y; z):
De…nition 13 The pair (X; d), where d is a metric on X, is called a metric
space.
Example 14 The natural metric on < is
d(x; y) = jx
yj :
There are several natural metrics on <M . The Euclidean metric is de…ned
by
s
M
P
(xi yi )2 :
d(x; y) =
i=1
The l1 metric (also called the taxi-cab metric for <2 ) is de…ned by
d(x; y) =
M
P
i=1
3
jxi
yi j :
The sup metric or uniform metric is de…ned by
d(x; y) = max jxi
i=1;:::;M
yi j :
Example 15 Let X be any set and de…ne metric d by d(x; x) = 0 and
d(x; y) = 1 if x 6= y: Then, d is a metric on X called the discrete metric.
Exercise Set 2
Exercise 4: Demonstrate that the metrics in the previous two examples
satisfy all of the four properties of a metric.
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Open and closed sets
De…nition 16 An (open) "-ball about x 2 E; where E is a set in a metric
space (X; d), is de…ned as
B" (x) = fy 2 E : d(y; x) < "g:
De…nition 17 A set E in a metric space (X; d) is open i¤ for each x 2 E
there exists " > 0 such that B" (x) E: A set is closed if its complement is
open.
De…nition 18 A point x 2 X is called a point of closure of set E in a
metric space (X; d) if 8 > 0 9y 2 E such that d(x; y) < : The set of points
of closure of E is denoted by E (or cl(E)):
The closure of a set E consists of all points which are intuitively “close
to E”.
Theorem 19 A set E is closed if and only if E = E:
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Frequently, condition E = E is used as a de…nition of a closed set.
The de…nition of a point of closure is closely related to the de…nition of
a limit point:
De…nition 20 A point x 2 X is called a limit point of set E in a metric
space (X; d) if 8 > 0 9y 2 E such that y 6= x and d(x; y) < :
The di¤erence between the two de…nitions is subtle but important —
namely, in the de…nition of the limit point, every open ball about point x
must contain a point of the set other than x itself.
Theorem 21 Every limit point is a point of closure, but not every point of
closure is a limit point.
De…nition 22 A point of closure which is not a limit point is called an
isolated point.
Theorem 23 The closure of a set E is equal to the union of E and the set
of limit points of E.
Example 24 In the natural metric d(x; y) = jx yj on <; the "-ball about
x 2 X is just the interval (x "; x + ") : This makes it clear that each interval
of the form (a; b) is an open set. There are obviously open sets of forms other
than (a; b) (Which are these forms?).
Theorem 25 The open sets in a metric space (X; d) have the following properties:
a) Any union of open sets is open;
b) Any …nite intersection of open sets is open;
c) ? and X are open.
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Given the de…nition of a closed set, the corresponding theorem for the
closed sets is:
Theorem 26 The closed sets in a metric space (X; d) have the following
properties:
a) Any intersection of closed sets is closed;
b) Any …nite union of closed sets is closed;
c) ? and X are closed.
De…nition 27 The interior Int(E) of set E in a metric space (X; d) is the
open set
Int(E) = fx 2 E : 9" > 0 such that d(y; x) < " and y 2 X imply y 2 Eg:
De…nition 28 The boundary Bdry(E) (also denoted @E) of set E in a
metric space (X; d) is the closed set
Bdry(E) = E n Int(E):
Every point of a set is either an interior point or a boundary point.
De…nition 29 A set E in a metric space (X; d) is bounded i¤ E
for some x 2 E and 0 < M < +1:
BM (x)
De…nition 30 A set E in the Euclidean space <M is compact i¤ it is closed
and bounded.
Exercise Set 3
Exercise 5: Prove Theorem 19.
Exercise 6: Prove Theorem 21.
Exercise 7: Prove Theorem 23.
Exercise 8: Prove Theorem 25.
Exercise 9: Prove that the interior of any set is open.
Exercise 10: Prove that the boundary of any set is closed.
Exercise 11: Prove that Int(E) = E
6
c c
:
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In…mum and Supremum
De…nition 31 The greatest lower bound (or in…mum, or inf) of a subset S of a partially ordered set (P; ); denoted as inf(S), is an element l of
P such that
1. l x for all x 2 S, and
2. For any p 2 P such that p x for all x 2 S it holds that p l.
De…nition 32 The least upper bound (or supremum, or sup) of a subset
S of a partially ordered set (P; ); denoted as sup(S); is an element u of P
such that
1. u x for all x 2 S, and
2. For any p 2 P such that p x for all x 2 S it holds that p u.
De…nition 33 (real numbers <) The in…mum of a set S
that
1. l x for all x 2 S, and
2. 8" > 0 9x 2 S such that l + " > x.
< is l 2 < such
De…nition 34 (real numbers <) The supremum of a set S
such that
1. u x for all x 2 S, and
2. 8" > 0 9x 2 S such that u " < x.
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< is u 2 <
Sequences
De…nition 35 By a sequence fxn g1
n=1 from a nonempty set X , we mean
an ordered set of the form (x1 ; x2 ; :::) where xi 2 X for all i 2 f1; 2; :::; 1g:
More formally, a sequence (xn )1
n=1 is a function from the set of natural numbers N into X. That is, sequence fxn g1
n=1 is the function f : N ! X such
that f (n) = xn for all n 2 N:
De…nition 36 Consider a sequence fxn g1
n=1 and a strictly increasing function : N ! N (that is, (k) < (l)for any k; l 2 N with k < l). The
1
sequence x (1) ; x (2) ; x (3) ; :::: = x (k) k=1 is called a subsequence of sequence fxn g1
n=1 .
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Example 37 Consider sequence (0; 1; 0; 1; 0; 1; :::): The subsequence formed
by taking the odd-numbered elements of the sequence is given by (0; 0; 0; :::):
The subsequence formed by taking the even-numbered elements of the sequence
is given by (1; 1; 1; :::):
De…nition 38 Let (X; d) be a metric space. A sequence fxn g1
n=1 converges
to x 2 X, written lim xn = x; i¤ d (xn ; x) converges to 0 as a sequence of
n!1
real numbers, i.e. lim d (xn ; x) = 0: Equivalently, lim xn = x i¤
n!1
n!1
8" > 0 9N 8n
N d (xn ; x) < ":
De…nition 39 (Real numbers R) A sequence of real numbers fxn g1
n=1 converges to x 2 R, written lim xn = x; i¤ jxn xj converges to 0 as a sequence
n!1
of real numbers, i.e. lim jxn
n!1
xj = 0: Equivalently, lim xn = x i¤
n!1
8" > 0 9N 8n
N jxn
xj < ":
De…nition 40 (<M with the Euclidean metric) A sequence fxn g1
n=1 from
2
M
1
2
M
<M (where xn = (x1n ; xs
;
:::;
x
))
converges
to
x
=
(x
;
x
;
:::;
x
) 2 <M ,
n
n
M
P
(xin xi )2 converges to 0 as a sequence of real
written lim xn = x; i¤
n!1
i=1
s
M
P
(xin xi )2 = 0: Equivalently, lim xn = x i¤
numbers, i.e. lim
n!1
8" > 0 9N 8n
6
n!1
i=1
N
s
M
P
i=1
(xin
xi )2 < ":
Continuous Functions
De…nition 41 If (X; ) and (Y; ) are metric spaces, a function f : X ! Y
is continuous at x0 2 X i¤ for each sequence fxn g1
n=1 that converges to x0 ;
lim f (xn ) = f (x0 ): Equivalently, f : X ! Y is continuous at x0 2 X i¤
n!1
8" > 0 9 > 0 8x 2 X [ (x; x0 ) <
8
! (f (x); f (x0 )) < "] :
De…nition 42 (Continuity for a real-valued function on < with the natural
metric on <) Let f : X ! Y; where X; Y
<. f : X ! Y is continuous at
x0 2 X i¤ for each sequence fxn g1
that
converges
to x0 ; lim f (xn ) = f (x0 ):
n=1
n!1
Equivalently, f : X ! Y is continuous at x0 2 X i¤
8" > 0 9 > 0 8x 2 X [jx
x0 j <
! jf (x)
f (x0 )j < "] :
We can rephrase the de…nition of a continuous function between metric
spaces in terms of open sets in these spaces, thus avoiding explicit mention
of the metrics involved:
Theorem 43 If (X; ) and (Y; ) are metric spaces, a function f : X ! Y
is continuous at x0 2 X i¤ for each open set set V in Y containing f (x0 )
there is an open set U in X containing x0 such that f (U ) V:
Thus, we can carry the de…nition of a continuous function to any setting
where we can carry a “reasonable” notion of an open set. A “reasonable”
de…nition of an open set is introduced in the following section, where we
introduce a mathematical construct that includes metric spaces as a special
case.
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Topological Spaces
De…nition 44 A topology on a set X is a collection
called the open sets, satisfying:
a) Any union of elements of belongs to ;
b) Any …nite intersection of elements of belongs to ;
c) ? and X belong to :
De…nition 45 The pair (X; ); where
logical space.
of subsets of X;
is a topology on X; is called a topo-
De…nition 46 Complements of open sets are called closed.
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De…nition 47 Let (X; d) be a metric space. Then, by Theorem 25, the open
sets de…ned by De…nition 17 form a topology called the metric topology d :
Whenever (X; ) is a topological space whose topology is the metric topology
d for some metric d on X; we call (X; d) a metrizable topological space.
Every metric space (X; d) de…nes a metrizable topological space (X; d ) and
given a metrizable topological space (X; ); one can always …nd many metrics
d on X such that d = : Two metrics generating the same topology are
equivalent.
The Euclidean, l1 , and sup metrics on <n are equivalent.
De…nition 48 A property of a metric space that can be expressed in terms
of open sets without mentioning a speci…c metric is called a topological
property.
There are many topological spaces that are not metrizable.
One can easily introduce a de…nition of convergence of a sequence in a
topological space as well as other notions that you are familiar with for metric
spaces. We only provide a de…nition of continuity of a function:
De…nition 49 If (X; ) and (Y; ) are topological spaces, a function f :
X ! Y is continuous at x0 2 X i¤ for each open set set V in Y containing
f (x0 ) there is an open set U in X containing x0 such that f (U ) V:
Note that this de…nition is similar to the one given for metric spaces.
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Lower and Upper Semicontinuity, and Continuity
De…nition 50 A real-valued function f : X ! < is upper semicontinuous if for each a 2 <, the upper contour set fx : f (x)
ag is closed (or
equivalently fx : f (x) > ag is open). It is lower semicontinuous if every
lower contour set fx : f (x) ag is closed (or equivalently fx : f (x) < ag is
open).
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The above de…nition is valid when X is a general (not necessarily metrizable) topological space.
Theorem 51 A real-valued function is continuous if and only if it is both
upper and lower semicontinuous.
Note that f is upper semicontinuous if and only if f is lower semicontinuous. We can even talk about semicontinuity at a point. We de…ne it
for the case when X is a metric space (One can easily introduce a similar
de…nition for a general topological space.)
De…nition 52 Let (X; d) be a metric space. The real-valued function f :
X ! < is upper semicontinuous at x 2 X i¤
f (x)
lim supf (y) = inf
sup
f (y):
">0 0<d(x;y)<"
y!x
Similarly, f is lower semicontinuous at x 2 X i¤
f (x)
lim inf f (y) = sup
y!x
inf
">0 0<d(x;y)<"
f (y):
Equivalently, f is upper semicontinuous at x 2 X i¤
8" > 0 9 > 0 8x 2 X [d (x; y) <
! f (y) < f (x) + "] :
Similarly, f is lower semicontinuous at x 2 X i¤
8" > 0 9 > 0 8x 2 X [d (x; y) <
9
! f (y) > f (x)
"] :
Correspondences, Lower and Upper Hemicontinuity
Let X and Y be topological spaces. If it is hard for you to think of X and
Y as topological spaces, think of them as metric spaces or a speci…c metric
space (for example, the Euclidean space).
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De…nition 53 A correspondence ' : X
Y associates to each point in X
a subset of Y . For a correspondence ', let
Gr(') = f(x; y) 2 X
Y : x 2 X; y = '(x)g
denote the graph of ':
De…nition 54 The upper (or strong) inverse of E under ' is de…ned by
'U (E) = fx 2 X : '(x)
Eg:
The lower (or weak) inverse of E under ' is de…ned by
'L (E) = fx 2 X : '(x) \ E 6= ?g:
De…nition 55 A correspondence ' is upper hemicontinuous (uhc) at x
if whenever x is in the upper inverse of an open set, so is a neighborhood of
x: That is, ' is uhc at x if 8open set E with x 2 'U (E) 9 open set A such
that x 2 A and A 'U (E):
' is lower hemicontinuous (lhc) at x if whenever x is in the lower inverse of an open set so is a neighborhood of x: That is, ' is lhc at x if 8open
set E with x 2 'L (E) 9 open set A such that x 2 A and A 'L (E):
The correspondence ' is upper hemicontinuous (respectively, lower hemicontinuous) if it is upper hemicontinuous (respectively, lower hemicontinuous) at
every x 2 X. Thus, ' is upper hemicontinuous (respectively, lower hemicontinuous) if the upper (respectively, lower hemicontinuous) inverses of open
sets are open.
A correspondence is continuous if it is both upper and lower hemicontinuous.’
When ' is compact-valued the above de…nitions of hemicontinuity are
equivalent to the following:
De…nition 56 A compact-valued correspondence ' is uhc at x if 8fxn g1
n=1
with lim xn = x and 8fyn g1
n=1 with yn 2 '(xn ) for all n 9 a convergent
n!1
1
subsequence fykn g1
n=1 of fyn gn=1 such that lim ykn 2 '(x):
n!1
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A compact-valued correspondence ' is lhc at x if 8y 2 '(x) and 8fxn g1
n=1
with lim xn = x 9N 1 and a sequence fyn g1
n=N such that lim yn = y and
n!1
yn 2 '(xn ) for all n
n!1
N:
If correspondence ' is singleton-valued, the upper and the lower inverses
of a set coincide and agree with the inverse regarded as a function. Either
form of hemicontinuity is equivalent to the continuity of a function. The term
“semicontinuity” has been used to mean hemicontinuity, but this usage can
lead to confusion when discussing real-valued singleton correspondences. A
semicontinuous real-valued function is not a hemicontinuous correspondence
unless it is also continuous.
De…nition 57 A correspondence ' is closed at x if 8fxn g1
n=1 with lim xn =
n!1
x and 8fyn g1
n=1 with yn 2 '(xn ) for all n; then y 2 '(x): A correspondence
is closed if it is closed at every point of its domain, that is, if its graph is
closed.
In general, a correspondence may be closed without being upper hemicontinuous, and vice versa. On the other hand, we have:
Theorem 58 Let X <m and Y
<k : Then
1. If ' is upper hemicontinuous and closed-valued, then ' is closed.
2. If Y is compact and ' is closed, then ' is upper hemicontinuous.
3. If ' is singleton-valued at x and upper hemicontinuous at x, then ' is
continuous at x:
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References:
1. Aliprantis, C.D. and K.C., Border, 1999. In…nite-Dimensional Analysis: A Hitchhiker’s Guide. Springer-Verlag, Berlin.
2. Kolmogorov, A.N. and S.V., Fomin, 1970. Introductory Real Analysis.
Dover Publications, New York.
3. Munkres, J.R., 2000. Topology. Prentice Hall.
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4. Royden, H.L., 1988. Real Analysis. Prentic Hall, New York.
5. Willard, S., 1970. General Topology. Dover Publications, New York.
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