examples of compact spaces

examples of compact spaces

Here are some

examples of compact spaces

(

http://planetmath.org/

Compact

):

•

The unit interval [0,1] is compact. This follows from the

Heine-Borel Theorem

. Proving that theorem is about as hard as proving directly that [0,1] is compact. The

half-open interval

(0,1] is not compact: the

open cover

(

1

/

n

,

1

]

for

n

=

1

,

2

,

…

does not have a finite subcover.

•

Again from the Heine-Borel Theorem, we see that the closed

unit ball

of any finite-dimensional

normed vector space

is compact. This is not true for infinite

dimensions

; in fact, a normed vector space is finite-dimensional if and only if its closed unit ball is compact.

•

Any finite topological space is compact.

•

Consider the set

2

ℕ

of all infinite

sequences

with entries in

{

0

,

1

}

. We can turn it into a

metric space

by defining

d

⁢

(

(

x

n

)

,

(

y

n

)

)

=

1

/

k

, where

k

is the smallest index such that

x

k

≠

y

k

(if there is no such index, then the two sequences are the same, and we define their distance to be zero). Then

2

ℕ

is a

compact space

, a consequence of

Tychonoff

’s theorem. In fact,

2

ℕ

is

homeomorphic

to the Cantor set (which is compact by Heine-Borel). This construction can be performed for any

finite set

, not just {0,1}.

•

Consider the set

K

of all functions

f

:

ℝ

→

[

0

,

1

]

and defined a

topology

on

K

so that a sequence

(

f

n

)

in

K

converges

towards

f

∈

K

if and only if

(

f

n

⁢

(

x

)

)

converges towards

f

⁢

(

x

)

for all

x

∈

ℝ

.
(There is only one such topology; it is called the topology of pointwise convergence). Then

K

is a compact

topological space

, again a consequence of Tychonoff’s theorem.

•

Take any set

X

, and define the

cofinite topology

on

X

by declaring a subset of

X

to be open if and only if it is empty or its

complement

is finite. Then

X

is a compact topological space.

•

The

prime spectrum

of any commutative ring with the

Zariski topology

is a compact space important in

algebraic geometry

. These prime spectra are almost never

Hausdorff spaces

.

•

If

H

is a

Hilbert space

and

A

:

H

→

H

is a continuous linear operator, then the

spectrum

of

A

is a compact subset of

ℂ

. If

H

is infinite-dimensional, then any compact subset of

ℂ

arises in this manner from some continuous linear operator

A

on

H

.

•

If

𝒜

is a complex C*-

algebra

which is

commutative

and contains a one, then the set

X

of all non-zero algebra homomorphisms

ϕ

:

𝒜

→

ℂ

carries a natural topology (the weak-* topology) which turns it into a compact Hausdorff space.

𝒜

is

isomorphic

to the C*-algebra of

continuous

complex-valued functions on

X

with the

supremum norm

.

•

Any

profinite group

is compact Hausdorff: finite

discrete spaces

are compact Hausdorff, therefore their

product

is compact Hausdorff, and a profinite group is a

closed subset

of such a product.

•

Any

locally compact Hausdorff space

can be turned into a compact space by adding a single point to it (

Alexandroff one-point compactification

(

http://planetmath.org/AlexandrovOnePointCompactification

)). The one-point compactification of

ℝ

is homeomorphic to the circle

S

1

; the one-point compactification of

ℝ

2

is homeomorphic to the sphere

S

2

. Using the one-point compactification, one can also easily

construct

compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.

•

Other non-Hausdorff compact spaces are given by the left

order topology

(or right order topology) on

bounded

totally ordered sets

.

Title

examples of compact spaces

Canonical name

ExamplesOfCompactSpaces

Date of creation

2013-03-22 12:48:47

Last modified on

2013-03-22 12:48:47

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

16

Author

yark (2760)

Entry type

Example

Classification

msc 54D30

Related topic

TopologicalSpace

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