5. Compact SetsJuly 12, 2019LetSbe a subset ofR. IfSis bounded and closed, then it is an important propertycalled: compactness. The definition of “compactness” looks bit strange but it is very useful
The definition of compact sets
Let
S
⊆
R
be a subset.
1. A collection
F
of some open sets in
R
such that all of open sets in
F
covers
S
, namely,
S
⊆
∪
A
∈F
A,
is called an
open cover
of
S
.
2. A subset
G
of
F
which also covers
S
is called a
subcover
of
S
. If the number of all
elements of
G
is finite,
G
is called a
finite subcover
of
S
.
3.
S
is
compact
if for every open cover
F
of
S
, it has a finite subcover of
S
, namely, there
is a finite open set
A
1
, A
2
, ..., A
n
as elements of
F
such that
1
S
⊆ ∪
n
j
=1
A
j
.
(1)
The historical origin
The above definition of compactness appears strange. Let us go
back history to find its trace.
At some time, central to the theory of analysis was the concept of uniform continuity
and a theorem stating that
Theorem 0.1
Every continuous function on a closed finite interval is uniformly continuous.
1
In short,
S
is compact if every open cover of
S
has a finite subcover.
41

2
T.H. Hildebrandt,
The Borel Theorem and Its Generalizations
, Bulletin of the AMS, vol. 32 (1926), pp.
423-474.
42

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- Topology, Metric space, Borel, Bolzano weierstrass theorem, set S