Open balls are open sets
Chapter 2: Basic Topology: Theorems
Open balls are open sets
What to show: We want to show that if you have an open ball and choose some point in that
ball, that you you can make an ball around that point that is completely contai
If p X is a limit point of a set E X, then every neighborhood of
p cointains innitely many points of E.
Chapter 2: Basic Topolgy: Theorems
What to show: We want to show that no matter how small of a radius you use to make a
ball around p, there will alway
Every innite subset of a compact set K has a limit point in K.
Chapter 2: Basic Topology: Theorems
What to show: That for every innte subset E belonging to a compact set, there is some
point x in the compact set such that r > 0(Br (x) x) E = .
Motivation
Every compact set is closed
Chapter 2: Basic Topology: Theorems
What to show: Two ways to look at this: rst, if K is closed, then it will contain all of its
limit points. Or, concersely, if we can show that K c is open, then K must be closed. This
entails
Connected Sets in R
Theorem: Let the set E R1 be non-empty. E is connected if and only if whenever x, y E
such that x < y and z such that x < z < y, then z E.
Motivation for proof:
Proof: Suppose E is not connected. Then there exist two non-empty sets A a
A set is open if and only its complement is closed
Chapter 2: Basic Topology: Theorems
E is open i Ec is closed.
What to show: Two things: rst, if a set is open then, its complement is closed; i.e. every
limit point of the complement belongs to the comple
Chapter 2: Basic Topolgy: Denitions
Let X be a metric space
Closed set: A set E is closed if it contains all of its limit points.
Compact set: A set K X is said to be compact if for any open cover cfw_G for I of
K there exists a nite subcover, i.e. G1 ,
Chapter 7: Series and Sequences of Functions
Denitions
Pointwise Convergence: We say the series of functions cfw_fn , n = 1, 2, . dened on a set
E converge pointwise on E to the limit function if
f (x) = lim fn (x) (x E)
n
(1)
or equivalently, > 0 there
Basic Topology
A set is open if and only if its complement is closed.
Neighborhoods of limit points contain innitely many points of the set.
Open balls or neighborhoods are open sets.
Intervals in R are connected.
Every innite and bounded subset of R
Denitions for Intermediate Analysis I
Absolute Convergence of a Series (Chapter 3): A series
an converges absolutely if
|an | converges.
Alternating Series: An alternating series has a form
n
0 (1) an
or
0
cos(n )an
where an > 0. Bounded function (Chapter