Universal space

In mathematics, a universal space is a certain metric space that contains all metric spaces whose dimension is bounded by some fixed constant. A similar definition exists in topological dynamics.

Definition

Given a class

C

{\displaystyle \textstyle {\mathcal {C}}}

of topological spaces,

U

∈

C

{\displaystyle \textstyle \mathbb {U} \in {\mathcal {C}}}

is universal for

C

{\displaystyle \textstyle {\mathcal {C}}}

if each member of

C

{\displaystyle \textstyle {\mathcal {C}}}

embeds in

U

{\displaystyle \textstyle \mathbb {U} }

. Menger stated and proved the case

d

=

1

{\displaystyle \textstyle d=1}

of the following theorem. The theorem in full generality was proved by Nöbeling.

Theorem:

The

(

2

d

+

1

)

{\displaystyle \textstyle (2d+1)}

-dimensional cube

[

0

,

1

]

2

d

+

1

{\displaystyle \textstyle [0,1]^{2d+1}}

is universal for the class of compact metric spaces whose Lebesgue covering dimension is less than

d

{\displaystyle \textstyle d}

.

Nöbeling went further and proved:

Theorem: The subspace of

[

0

,

1

]

2

d

+

1

{\displaystyle \textstyle [0,1]^{2d+1}}

consisting of the set of points, at most

d

{\displaystyle \textstyle d}

of whose coordinates are rational, is universal for the class of separable metric spaces whose Lebesgue covering dimension is less than

d

{\displaystyle \textstyle d}

.

The last theorem was generalized by Lipscomb to the class of metric spaces of weight

α

{\displaystyle \textstyle \alpha }

,

α

>

ℵ

0

{\displaystyle \textstyle \alpha >\aleph _{0}}

: There exist a one-dimensional metric space

J

α

{\displaystyle \textstyle J_{\alpha }}

such that the subspace of

J

α

2

d

+

1

{\displaystyle \textstyle J_{\alpha }^{2d+1}}

consisting of set of points, at most

d

{\displaystyle \textstyle d}

of whose coordinates are "rational" (suitably defined), is universal for the class of metric spaces whose Lebesgue covering dimension is less than

d

{\displaystyle \textstyle d}

and whose weight is less than

α

{\displaystyle \textstyle \alpha }

.

Universal spaces in topological dynamics

Consider the category of topological dynamical systems

(

X

,

T

)

{\displaystyle \textstyle (X,T)}

consisting of a compact metric space

X

{\displaystyle \textstyle X}

and a homeomorphism

T

:

X

→

X

{\displaystyle \textstyle T:X\rightarrow X}

. The topological dynamical system

(

X

,

T

)

{\displaystyle \textstyle (X,T)}

is called minimal if it has no proper non-empty closed

T

{\displaystyle \textstyle T}

-invariant subsets.