February 6, 2006
Topology 440, Homework no. 1
(Due on Wednesday, February 15)
1. Let X be the set X = cfw_1, 2, 3. List all possible topologies on X. How many are
there?
2. Let X be an innite set and p X some arbitrary element in X. Let TF,p be the
collec
1. The one point compactification
Denition 1.1. A compactication of a topological space X is a compact topological
space Y containing X as a subspace.
Given any non-compact space X, compactications always exist. This section explores the smallest possible
CONTINUITY ON Rn VIA OPEN SETS
1. Continuity and convergence
Recall that the distance function dist : Rn Rn R is dened as
dist(x, y) =
(x1 y1 )2 + . + (xn yn )2
where xi and yi are the coordinates of x and y. We dene Bx (r) to be the ball of radius
r > 0
April 24, 2006
Topology 440, Homework no. 4
(Due on Monday, May 1)
1. Let X be a second countable topological space and let C be any open cover of X.
Show that there is a countable cover D = cfw_D1 , D2 , D3 , . with the property that for
each Di D there
EXAMPLES OF TOPOLOGICAL SPACES
1. Metric spaces
Denition 1.1. Let X be a nonempty set and let d : X X [0, be a function
subject to the conditions
1. d(x, y) = 0 if and only if x = y.
2. d(x, y) = d(y, x) (symmetry).
3. d(x, z) d(x, y) + d(y, z) (triangle
April 5, 2006
Topology 440, Homework no. 3
(Due on Monday, April 17)
1. Let : X Y be a quotient map. Show that if Y is connected and for each y Y
the subspace 1 (cfw_y) is connected then X is connected as well.
2. Let X be a topological space with the pro
Convergent sequences in topological spaces
1. Definition and examples
Denition 1.1. Let (X, T ) be a topological space and xn X a sequence. We say that
the sequence xn converges to x0 X if for every open set U X which contains x0
there exists an n0 N such
1. One dimensional manifolds
The goal of this note is to give a complete list, up to homeomorphism, of 1-manifolds
which are connected and Hausdor . Dropping the connectedness condition one only obtains
(disjoint) unions of connected ones. However, droppi
February 6, 2006
Topology 440, Homework no. 2
(Due on Wednesday, February 22)
1. Let X be a topological space and A a subset of X. Show that
X A = X Int(A)
Int(X A) = X A
and
Use these equalities to show that A = (X A).
2. Let (X, <) be a simply ordered s
1. Manifolds with boundary
Denition 1.1. An n-dimensional manifold with boundary is a topological space X in which
each point x X has a neighborhood Ux homeomorphic to either Rn or to H n the closed
upper half-space in Rn :
H n = cfw_(x1 , ., xn ) Rn | xn
1. Compactness for metric spaces
For a metric space (X, d) we will as usual use the notation Bx (r) to denote the open
ball of radius r centered at x X:
Bx (r) = cfw_y X | d(x, y) < r
Denition 1.1. We say that a metric space (X, d) is totally bounded if f
MANIFOLDS
1. Introduction
Denition 1.1. A topological n-dimensional manifold (or n-manifold for short) is a
topological space X in which each point x X has a neighborhood Ux which is homeomorphic
to Rn equipped with the Euclidean topology. A chart around
Separation axioms
1. The axioms
The following categorization of topological spaces below is a measure of how neor
how coarsethe topology of the space is.
1. A topological space is said to be T0 or Kolmogorov if for any two points
x, y X there exists an op