Chapter 5 Paracompactness
Let cfw_W I be a cover of X . (We do not assume W is open.) Denition 5.0.27 A cover cfw_T J is called a renement of cfw_W I if J, I s.t. T W . Denition 5.0.28 A collection cfw_W I of subsets of X is called locally nite if each x

Chapter 6 Connectedness
Denition 6.0.40 A pair of nonempty open subsets A and B of a topological space X is called a disconnection of X if A B = and A B = X . Note: If A, B is a disconnection of X then A and B are also closed since A = B c and B = Ac . Pr

SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
Problem Set I (Solution Sketch)
MAT C27F
1. Since A is closed in Y , a set B which is closed in X such that A = B Y .
Therefore A is the intersection of two closed sets in X , and so it is closed in X .
2.
a) Let h

SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
Problem Set III (Solution Sketch)
MAT C27F
1. To show X is homeomorphic to S 1 S 1 :
The canonical projection q : [0, 1] [0, 1] X is surjective.
[0, 1] [0, 1] is compact, since it is a closed bounded subset of R3 .

1. Dene f : X S 1 by f (t) = (cos 2t, sin 2t)
[0, 1]
(cos 2t, sin 2t)
q
S1
f
X
If q : [0, 1] X is the quotient map (taking each element to its equivalence class) then f q is the continous function t (cos 2t, sin 2t)
on [0, 1].
It follows that f is continu

Chapter 2
Topological Spaces
2.1
Metric spaces
Denition 2.1.1 A metric space consists of a set X together with a function d : X X R +
s.t.
1. d(x, y) = 0 x = y
2. d(x, y) = d(y, x)
x, y
3. d(x, z) d(x, y) + d(y, z)
x, y, z
triangle inequality
Example 2.1.

Chapter 3
Separation axioms
Let X be a topological space.
Denition 3.0.12 X has the following names if it has the following properties:
1. X is T0 if x = y X either open U s.t. x U, y U x U, y U or open
/
/
U s.t. x U, y U
/
2. X is T1 if x = y X open U s

SCARBOROUGH CAMPUS
UNIVERSITY OF TORONTO
Problem Set IV (Solution Sketch)
MAT C27F
1. Suppose X R is connected.
Let x, y, belong to X and assume x < z < y.
If z X then (, z) X and (z, ) X forms a disconnection of X, contradicting
X connected.
Therefore x

Chapter 2 Compactness
Denition 2.0.7 A topological space X is called compact if it has the property that every open cover of X has a nite subcover. Theorem 2.0.8 Heine-Borel A subset X Rn is closed and bounded if and only if every open cover of X has a ni