In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which for every sequence of open covers
U
1
,
U
2
,
…
{\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots }
of the space there are finite sets
F
1
⊂
U
1
,
F
2
⊂
U
2
,
…
{\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subset {\mathcal {U}}_{2},\ldots }
such that the family
F
1
∪
F
2
∪
⋯
{\displaystyle {\mathcal {F}}_{1}\cup {\mathcal {F}}_{2}\cup \cdots }
covers the space.
Contents
History
In 1924, Karl Menger
introduced the following basis property for metric spaces:
Every basis of the topology contains a countable family of sets with vanishing
diameters that covers the space. Soon thereafter,
Witold Hurewicz
observed that Menger's basis property can be reformulated to the above form using sequences of open covers.
Menger's conjecture
Menger conjectured that in ZFC every Menger metric space is σ-compact.
A. W. Miller and D. H. Fremlin
proved that Menger's conjecture is false, by showing that there is,
in ZFC, a set of real numbers that is Menger but not σ-compact.
The Fremlin-Miller proof was dichotomic, and the set witnessing the failure
of the conjecture heavily depends on whether a certain (undecidable) axiom
holds or not.
Bartoszyński and Tsaban
gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact.
Combinatorial characterization
For subsets of the real line, the Menger property can be characterized using continuous functions into the Baire space
N
N
{\displaystyle \mathbb {N} ^{\mathbb {N} }}
.
For functions
f
,
g
∈
N
N
{\displaystyle f,g\in \mathbb {N} ^{\mathbb {N} }}
, write
f
≤
∗
g
{\displaystyle f\leq ^{*}g}
if
f
(
n
)
≤
g
(
n
)
{\displaystyle f(n)\leq g(n)}
for all but finitely many natural numbers
Properties
Every compact, and even σ-compact, space is Menger.
Every Menger space is a Lindelöf space
Continuous image of a Menger space is Menger
The Menger property is closed under taking
F
σ
{\displaystyle F_{\sigma }}
subsets
Menger's property characterizes filters whose Mathias forcing notion does not add dominating functions.