1701354 Introduction to Topology – Nguyen
2-14-06
Class Handout
CURVES IN THE PLANE
I. Continuity for Vector Functions from
to
2
:
Definition: A mapping f : D   2 defined by f (t )  ( f1 (t ), f 2 (t )) in terms of its
component functions f1 , f 2 is called a vector function.
Example: Now that we know how to measure distance on 2 using the formula defined by 
(called a metric), it is straightforward to define continuity for vector functions. Write a
definition for a function f : D   2 to be continuous at a point a  D :
Definition A2:
Example: Let f : (0, 2)  2 be defined by f (t )  (3t ,1/ t ) :
(a) Graph the image of f.
(b) Prove that f is continuous at a  1 if and only if each of its component functions,
f1 (t )  3t and f 2 (t )  1/ t , are continuous at a  1 by using    arguments.
The example above reveals the following general connection between vector functions and their
components in terms of continuity:
Theorem: A function f : D   2 is continuous at a  D if and only if each of its
component functions f1 , f 2 : D  where f (t )  ( f1 (t ), f 2 (t )) are continuous at a  D .
Sequential Continuity
Just as before, continuity can also be described in terms of sequences:
Theorem: A function f : D   2 is continuous at a  D if and only if for every sequence
{xn } that converges to a, we have f ( xn )  f (a) .
II. Curves
Definition: A plane curve  is a continuous map  :[a, b]  2 . The image of  is denoted by
*. The initial and end points are given by  (a) and  (b) , respectively. If  (a)   (b) , then the
curve is said to be closed.
Space-Filling Curves
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1701354 Introduction to Topology – Nguyen
2-14-06
It is naïve to assume that the continuous image of a one-dimensional domain must be onedimensional as the following theorem demonstrates otherwise.
Theorem: (Peano) There exist a curve space-filling curve  :[0,1] 
the entire unit square:
 *  [0,1]  [0,1]  {( x, y)  2 : x, y [0,1]}
2
such that its image is
This space-filling curve is called the Hilbert curve. Its construction depends first on defining
special subintervals and sub-squares.
Part A. Construction of Hilbert Subintervals and Sub-squares
Consider the sequence of collections of subsets constructed in stages:
Stage 1:
Division of [0,1] into 4 closed subintervals (arranged in increasing order):
{I 0 , I1 , I 2 , I3}  {[0,1/ 4],[1/ 4,1/ 2],[1/ 2,3/ 4],[3/ 4,1]}
Division of [0,1]  [0,1] into 4 closed sub-squares (arranged counter-clockwise):
{S0 , S1 , S2 , S3}  {[0,1/ 2]  [0,1/ 2],[1/ 2,1]  [0,1/ 2],
[1/ 2,1]  [1/ 2,1],[0,1/ 2]  [1/ 2,1]}
Stage 2:
Division of [0,1] into 16 closed subintervals (based on Stage 1):
{I 00 , I 01 , I 02 , I 03 ,..., I 30 , I 31 , I 32 , I 33}
Division of [0,1]  [0,1] into 16 closed sub-squares (based on Stage 1):
{S00 , S01 , S02 , S03 ,..., S30 , S31 , S32 , S33}
Stage n:
Division of [0,1] into 4 n closed subintervals (based on Stage n-1):
{I i1i2 ...in }, i1i2 ...in {0,1, 2,3}
Division of [0,1]  [0,1] into 4 n closed sub-squares (based on Stage n-1):
{Si1i2 ...in }, i1i2 ...in  {0,1, 2,3}
Nested Property of Hilbert Subintervals and Sub-squares
I i1  I i1i2  I i1i2i3  ...
Si1  Si1i2  Si1i2i3  ...
Base Decimal Representation of Real Numbers in [0,1]
Base 10: Let {d n } be a sequence of digits such that dn {0,1, 2,...,9} . We define the base 10
decimal expansion of a real number x having digits d n as
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1701354 Introduction to Topology – Nguyen
2-14-06
def
x  0.d1d 2 d3 ...  0.d1  0.0d 2  0.00d3  ...  d1  0.1  d 2  0.01  d3  0.001  ...
n
d
d
d1 d 2
 2  33  ...  lim  ii
n 
10 10 10
i 1 10
Base 4: Let {d n } be such that dn {0,1, 2,3} . Then we define the base 4 decimal expansion of x
with digits d n as

def
x  0.d1d 2 d3 ... 
n
d
d1 d 2 d3
 2  3  ...  lim  ii
n

4 4
4
i 1 4
Example:
(a) Find a base 4 decimal expansion of the following values of x: 0, 1, and 1/2. Is the
decimal expansion of a given number unique?
(b) Prove that if t  [0,1] , then t belongs to infinitely many Hilbert subintervals, i.e. there
exists a sequence of digits {an } with an {0,1, 2,3} and such that t  I a1a2 ...an for all n.
Moreover, prove that t has base 4 decimal expansion whose digits are an , i.e.
t  0.a1a2 a3 ... .
Part B. Construction of Map  (to be explained on the next handout)
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