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 R
2
, an open disk is a set of form
{(x,y): (x-a)
2
+ (y-b)
2
< r}
ie) an integer of circle.
R
2
, A
subset S of R
2
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 > x
2
}
Note: (in R and R
2
): the union of any number of open sets is open.
Eg. Let In = (-1/n, 1/n)
∞
∩
In = {0} not open.
n=1
The intersection of open sets need not be open.
The intersection of 2 open sets is open. Let S
1
,S
2
be open and suppose x
∈
S
1
∩
S
2
there exists,
(a
1
,b
1
) such that x
∈
(a
1
,b
1
) & (a
1
,b
1
)
⊂
S
1
there exists (a
2
,b
2
) such that x
∈
(a
2
,b
2
) & (a
2
,b
2
)
⊂
S
2
a
1
a
2
x
b
1
b
2
(a
2
,b
1
)
⊂
S
1
∩
S
2
(S
1
∩
S
2
)
∩
S
3
We define
∅
to be open.