Defn Suppose (X,d) is a metric space and A is a subset of X.
Proposition A set O in a metric space is open if and only if each of its points are interior points.
Proposition A set C in a metric space is closed if and only if it contains all its limit points.
Defn Suppose (X,d) is a metric space and A is a subset of X. The closure of A is the smallest closed subset of X which contains A. The derived set A' of A is the set of all limit points of A.
Proposition The closure of A may be determined by either
or
Sequential Convergence
Defn
A sequence {xn} in a metric space (X,d) is said to converge,
to a point x0 say, if for each neighborhood of x0 there
exists a natural number N so that xn belongs to the neighborhood
if n is greater or equal to N; that is, eventually
the sequence is contained in the neighborhood.
In this case, we say that x0 is
the limit of the sequence
and write
xn := x0 .
Proposition In a metric space, sequential limits are unique.
Proposition That
a sequence {xn} converges in
a metric space (X,d) to a point x0 is
equivalent to the condition that for each
> 0 there is a natural
number N such that N
n
implies d(xn , x0)
<
.
Examples
Defn
A function f defined
on X\{x0}, with values in a metric space {Y,d2} is
said to have a limit L at x0 if
x0 is a limit point of X and for each neighborhood O2 of
L, there is a neighborhood O1 of x0 such that f maps
each element of the deleted neighborhood O1\{x0}
into O2 . This is denoted
f(x) := L.
Homework
This is equivalent to the condition: for each >
0 there is a
> 0 such that if 0 <
d1(x,x0) <
,
then d2(f(x),L) <
.
Proposition A necessary and sufficient condition for a function f to have a limit L at x0 is that for each sequence {xn} which converges to x0 (no point of which is equal to x0), then {f(xn)} converges to L. Consequently, if a function has a limit at a point x0, then it is unique.
Defn A function f is called continuous at a point x0 if either
Homework
A necessary and sufficient condition
for a function f to be continuous at x0 is
that for each >
0 there is a
> 0 such that if d1(x,x0)
<
,
then d2(f(x),f(x0))
<
.
Continuity
Defn Suppose
f : X Y where (X,d1)
and (Y,d2) are metric spaces. f is
called continuous
if the inverse image of each
open set in Y is open in X.
Proposition A
function f : X Y
is continuous if and only if the inverse image of each closed set in Y
is closed in X.
Theorem A
function f : X Y
is continuous if and only if f is continuous at each point of X.
Theorem Suppose
that f: X Y and g: Y
Z
are continuous functions, then gof
is a continuous function from X to Z.
Theorem Suppose that (X,dX) and (Y,dY) are both metric spaces, then X x Y is a metric space if the metric d is defined for zi = (xi,yi), i=1,2, by
d(z1,z2) := dX(x1,x2) + dY(y1,y2).
Examples: