Download Topology (Part 1) - Department of Mathematics, University of Toronto

Topology (Part 1)
Original Notes adopted from April 2, 2002 (Week 25)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
R, A subset S of R is open if S is a union of open intervals;
ie) whenever x ∈ S , there exists open interval (a,b) such that x ∈ (a,b) & (a,b) ⊂ S.
7
-2.1
3
7
9
13
17
S = { x: x< -7 or x ∈ (-2.1, 3) or x ∈ (7,9) or x ∈ (13,17)}
In R2, an open disk is a set of form
{(x,y): (x-a) 2 + (y-b) 2 < r}
ie) an integer of circle.
R2, A subset S of R2 is open if S is a union of open disks;
ie) whenever (x,y) ∈ S , there exists open disk D such that x ∈ D & D ⊂ S.
eg. {(x,y): y > x2 }
Note: (in R and R2 ): the union of any number of open sets is open.
Eg. Let In = (-1/n, 1/n)
’
@,Q = {0} not open.
n=1
The intersection of open sets need not be open.
The intersection of 2 open sets is open. Let S1 ,S2 be open and suppose x ∈ S1 @6 2 there exists,
(a1 ,b1 ) such that x ∈ (a1 ,b1 ) & (a1 ,b1 ) ⊂ S1
there exists (a2 ,b2 ) such that x ∈ (a2 ,b2 ) & (a2 ,b2 ) ⊂ S2
a1 a2
x
b1
(a2 ,b1 ) ⊂ S1 @6 2
(S1 @6 2) @6 3
We define ∅ to be open.
b2
For R & R2 , we have:
1) ∅ & whole space are open.
2) The union of any number of open sets is open
3) The intersection of any finite number of open sets is open.
A topological space is a set together with a collection of surds, called the open sets, such that the union of
any number of open sets is open, the intersection of a finite number of open sets is open, and the empty set
and the entire space are both open.
EXAMPLES OF TOPOLOGICAL SPACES.
1) R , R2 with their usual topologies, as above.
2) Let X = {a,b,c} and let every subset of X be designated as "open".
3) Take set R and designate a subset S as "open" if R\S (complement) is countable or if S = ∅.
S1 , S2 .... R\ S1 ,R\ S2 countable R\ S1 U R\ S2
Let x be any set with subsets designated "open" satisfying 3 properties, x is a topological space.
Definition: A subset F is closed if X\F is open.
Definition: A point x is a limit point of a subset S if whenever U is an open set containing x, U
contains a point of S other than x (x may or may not be in S).
Theorem: A set is closed if and only if it contains all its limit points.
Proof: Suppose S is closed. Then X\S is open. If x ∈ X\S , then x is not a limit point since X\S is an open
set whose intersection with S is empty.
∴ all limit points are in S, not X\S.
Conversely, (if it contains all its limit points its closed). Suppose S contains all its limit points.
Show: X\S open. Let x ∈ X\S x not a limit point of S, so there exists Ux , Ux @ S = ∅.
Ux ⊂ X \S.
X\S = U Ux
x∈ X\S
X\S is a union of open sets, so X\S is open
Definition: If f: X→
→ Y, and S ⊂ Y, then f--1(S) = {x ∈ X: f(x) ∈ S}
("The inverse image of S").
Definition: If X and Y are topological spaces, & f: X →Y, then f is continuous if f- -1(U) is an open
subset of X whenever U is an open subset of Y.
Note: For f: R →R, above agrees with usual definition.
∀ ∈>0, there exists δ >0 such that |f (x) – f(a) | < ∈ whenever |x-a| < δ
_
Definition: The closure of a subset S of a topological space is S U {limit points of S} denoted by S.
Eg. In R with usual topology. _
1) If S = (3,9),
S = [3,9]
2)S = {1/n: n =1,2,3 ...}
0 is a limit point.
1 = S U {0}
Definition: Homeomorphism is a one-to-one function from a topological space onto another
topological space which is continuous and which has a continuous inverse. Two topological spaces
are said to be homeomorphic if there is a homemorphism from one onto the other.
Examples:
[-Homeomorphic]
[Not Homeomorphic]
Disconnected ______ _______
Definition: The topological space X is disconnected if there exists subsets A & B of X both of which are
closed, neither of which is empty, such that
X = AUB & A @% ;
Then A & B are both open too.
In every topological space, X& Y are both open and closed.
A space is disconnected if and only if it has a subset 0 in addition to
∅ & x that is both open & closed.