COMPACTNESS VS. SEQUENTIAL COMPACTNESS
The aim of this handout is to provide a detailed proof of the equivalence between the two
definitions of compactness: existence of a finite subcover of any open cover, and existence of a limit
point of any infinite subset (also called in class
sequentially compactness
).
Definition 1.
K
is
compact
if every open cover of
K
contains a finite subcover.
K
is
sequentially
compact
if every infinite subset of
K
has a limit point in
K
.
Theorem 1.
K
is compact
⇐⇒
K
is sequentially compact.
The first half of this statement (compact =
⇒
sequentially compact) is Theorem 2.37 in Rudin and
is proved there. Our aim is to prove the converse implication (sequentially compact =
⇒
compact),
following the lines of Exercises 23, 24 and 26 in Rudin Chapter 2.
The proof requires the introduction of two auxiliary notions:
Definition 2.
A space
X
is
separable
if it admits a countable dense subset.
For example
R
is separable (
Q
is countable, and it is dense since every real number is a limit of
rationals); for the same reason
R
k
is separable (consider all points with only rational coordinates).
Definition 3.
A collection
{
V
α
}
of open subsets of
X
is said to be a
base
for
X
if the following
is true: for every
x
∈
X
and for every open set
G
⊂
X
such that
x
∈
G
, there exists
α
such that
x
∈
V
α
⊂
G
.
In other words, every open subset of
X
decomposes as a union of a subcollection of the
V
α
’s – the
V
α
’s “generate” all open subsets. The family
{
V
α
}
almost always contains infinitely many members
(the only exception is if
X
is finite). However, if
X
happens to be separable, then countably many
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 Fall '10
 Prof.KatrinWehrheim
 Topology, Metric space, Vα, General topology, FN, limit point

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