Topology of
Definition T.1: Let x  . If   0 the interval S  ( x   , x   ) is called an  -neighborhood
(or simply a neighborhood) of x.
A deleted neighborhood of x is a set of the form { y : 0  y  x   } .
Definition T.2: A set S  is said to be open if and only if for each x  S there exists a
neighborhood, N, of x such that N  S ,
Exercises:
Determine, with proof, which of the following sets are open:
1. (0,1)
2. {3,5}
3.
n
4.
X i where X i is open for each i
i 1

5.
X i where X i is open for each i
i 1
(continued) Determine, with proof, which of the following sets are open:
n
6.
X i where X i is open for each i
i 1

7.
X i where X i is open for each i
i 1
Definition T.3: A point p is an accumulation point (or limit point) of the set S  if and
only if for each  > 0, the interval  p   , p    contains at least one point of S other
than p.
Definition T.4: A set A is said to be closed if and only if it contains all of its accumulation
points.
Exercises:
Determine, with proof, which of the following sets are closed:
1

8.  : n  
n

(continued) Determine, with proof, which of the following sets are closed:

  3
9. sin x : x    ,
 2 2




10. The Cantor Middle-thirds Set
n
11.
X i where X i is closed for each i
i 1

12.
X i where X i is closed for each i
i 1
n
13.
X i where X i is closed for each i
i 1

14.
X i where X i is closed for each i
i 1
More Exercises:
15. Give an example (if possible) of a set S 
, such that S is neither open nor closed.
16. Give an example (if possible) of a set S 
, such that S is both open and closed.
Theorem T.5: A set S 
is open if and only its complement is closed.
We leave the proof as an exercise for the reader.
Definition T.6: Two sets are separated if and only if they are disjoint and neither contains
a limit point of the other.
Exercises: Determine whether the following sets are separated:
1. Rationals, Irrationals.
2. Cantor Set, Complement of Cantor Set

3. Integers, {x : x  ( z   , z  ), where z 
2
Definition T.7: A set S 
non-empty separated sets.
and   (0,1)
is connected if and only if it cannot be partitioned into two
Make up a theorem of your own – proof required.
Insert name
’s Theorem: A set S 
is connected if and only
Definition T.8: a collection of open sets Oi iA is said to cover a set S 
S
iA
if and only if
Oi . In this case, Oi iA is said to be an open cover of S. If B  A and Oi iB is
also an open cover of S, then Oi iB is called a open sub-cover of Oi iA . In particular,
if B is finite, Oi iB is called a finite sub-cover.
Definition T.9: A set S 
open sub-cover.
is compact if and only if every open cover of S has a finite
Exercises: Determine which sets are compact:
1. [0,1]
2. (0,1)
3.
Theorem T.10 (Heine – Borel): A set S 
and bounded.
Proof: Left as an exercise.
is compact if and only if S is both closed