Selection principle

This article is not about the anthropic principle.

In mathematics, the theory of selection principles deals with the possibility of obtaining mathematically significant objects by selecting elements from sequences of sets. The studied properties mainly include covering properties, measure- and category-theoretic properties, and local properties in topological spaces, especially functions spaces. Often, the characterization of a mathematical property using a selection principle is a nontrivial task leading to new insights on the characterized property.

Background and definitions

In 1924, Karl Menger ^[1] 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 ^[2] observed that Menger's basis property can be reformulated as the following selective property: for every sequence of open covers of the space, one can choose finitely many sets from each given cover, such that the family of chosen sets covers the space.

Hurewicz's reformulation of Menger's property was the first of several important topological properties described by a selection principle. Topological spaces having this covering property are nowadays called Menger spaces.

Let $\mathbf {A}$ and $\mathbf {B}$ be classes of mathematical objects. In 1996, Marion Scheepers ^[3] introduced the following selection hypotheses, that capture a large number of classic mathematical properties:

${\text{S}}_{1}(\mathbf {A} ,\mathbf {B} )$ : for every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots \in \mathbf {A}$ there are $U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\dots$ such that $\{U_{n}:n\in \mathbb {N} \}\in \mathbf {B}$ .
${\text{S}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )$ : for every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots \in \mathbf {A}$ there are finite sets ${\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots$ such that $\bigcup _{n}{\mathcal {F}}_{n}\in \mathbf {B}$ .
${\text{U}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )$ : for every sequence ${\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots \in \mathbf {A}$ , none containing a finite subcover, there are finite ${\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots$ such that $\{\bigcup {\mathcal {F}}_{n}:n\in \mathbb {N} \}\in \mathbf {B}$ .

The principles ${\text{S}}_{1}$ , ${\text{S}}_{\text{fin}}$ , and ${\text{U}}_{\text{fin}}$ are examples of selection principles. Later, Boaz Tsaban identified the prevalence of the following related principle:

${\binom {\mathbf {A} }{\mathbf {B} }}$ : Every member of $\mathbf {A}$ contains a member of $\mathbf {B}$ .

Covering properties

Let $X$ be a topological space. Denote by $\mathbf {O}$ the collection of all open covers of the space $X$ . (For technical reasons, we also request that the entire space $X$ itself is not a member of the cover.)

A topological space is a Menger space if it satisfies ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ .
A topological space is a Rothberger space if it satisfies ${\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )$ .

An infinite open cover ${\mathcal {U}}$ of $X$ is called a $\gamma$ -cover if every $x\in X$ belongs to all but finitely many sets $U\in {\mathcal {U}}$ . Denote the collection of $\gamma$ -covers of $X$ by $\mathbf {\Gamma }$ .

A topological space is a Hurewicz space if it satisfies ${\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )$ .

An open cover ${\mathcal {U}}$ of $X$ is called an $\omega$ -cover if every finite subset of $X$ is contained in some member of ${\mathcal {U}}$ . Denote the collection of $\omega$ -covers of $X$ by $\mathbf{\Omega}$ .

A topological space is a γ-space if it satisfies ${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}$ .

The above properties are the most celebrated selection principles.

Other properties

Using selection principles, we can express also non-covering properties.

Let $Y$ be a topological space, and $y\in Y$ . Denote by $\mathbf {\Omega _{y}}$ the family of subsets $A$ of the space $Y$ such that $y\in {\overline {A}}$ . By $\mathbf {\Omega _{y}^{\text{ctbl}}}$ we mean the family $\mathbf {\Omega _{y}}$ with the restriction to countable subsets. Denote by $\mathbf {\Gamma _{y}}$ the family of sequences in $Y$ that converge to $y$ .

A space $Y$ is Fréchet–Urysohn iff it satisfies ${\binom {\mathbf {\Omega _{y}} }{\mathbf {\Gamma _{y}} }}$ for all points $y\in Y$
A space $Y$ is strongly Fréchet–Urysohn iff it satisfies ${\text{S}}_{1}(\mathbf {\Omega _{y}} ,\mathbf {\Gamma _{y}} )$ for all points $y\in Y$
A space $Y$ has countable tightness iff it satisfies ${\binom {\mathbf {\Omega _{y}} }{\mathbf {\Omega _{y}^{\text{ctbl}}} }}$ for all points $y\in Y$
A space $Y$ has countable fan tightness iff it satisfies ${\text{S}}_{\text{fin}}(\mathbf {\Omega _{y}} ,\mathbf {\Omega _{y}} )$ for all points $y\in Y$
A space $Y$ has countable strong fan tightness iff it satisfies ${\text{S}}_{1}(\mathbf {\Omega _{y}} ,\mathbf {\Omega _{y}} )$ for all points $y\in Y$

Topological Games

Often there are close connections between Selection Principles and Topological Games. For example ${\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ is equivalent to the first player (Alice) not having a winning strategy in the game ${\text{G}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$ . Furthermore, in the previous equivalence we could replace the subscript fin in both cases by 1. Indeed, ${\text{S}}_{\text{1}}(\mathbf {A} ,\mathbf {B} )$ is equivalent to Alice not having a winning strategy in the game ${\text{G}}_{\text{1}}(\mathbf {A} ,\mathbf {B} )$ also for $({\mathbf {A}},{\mathbf {B}})$ any of $(\mathbf {O} ,\mathbf {O} )$ , $(\mathbf {\Omega } ,\mathbf {\Gamma } )$ , $(\mathbf {\Omega } ,\mathbf {\Omega } )$ and $(\mathbf {\Gamma } ,\mathbf {\Gamma } )$ ^[4]

Examples and properties

Every Menger space is a Lindelöf space
Every σ-compact space (a countable union of compact spaces) is Hurewicz
${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}\Rightarrow {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )\Rightarrow {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$
${\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}\Rightarrow {\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )\Rightarrow {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )$
Consistently, any of the above implications can not be reversed in the class of subsets of the real line.
Every Luzin set is Menger but no Hurewicz
Every Sierpiński set is Hurewicz

Subsets of the real line $\mathbb {R}$ (with the induced subspace topology) holding selection principles properties, most notably Menger and Hurewicz spaces, can be characterized by their continuous images in the Baire space ${\mathbb {N}}^{{\mathbb {N}}}$ . For functions $f,g\in \mathbb {N} ^{\mathbb {N} }$ , write $f\leq ^{*}g$ if $f(n)\leq g(n)$ for all but finitely many natural numbers $n$ . Let $A$ be a subset of ${\mathbb {N}}^{{\mathbb {N}}}$ . The set $A$ is bounded if there is a function $g\in \mathbb {N} ^{\mathbb {N} }$ such that $f\leq ^{*}g$ for all functions $f\in A$ . The set $A$ is dominating if for each function $f\in \mathbb {N} ^{\mathbb {N} }$ there is a function $g\in A$ such that $f\leq ^{*}g$ .

A subset of the real line is Menger iff every continuous image of that space into the Baire space is not dominating.
A subset of the real line is Hurewicz iff ever continuous image of that space into the Baire space is bounded.

Interconnections to other fields

General topology

Every Menger space is a D-space

Let P be a property of spaces. A space $X$ is productively P if, for each space $Y$ with property P, the product space $X\times Y$ has property P.

Every separable productively paracompact space is Hurewicz.
Assuming the Continuum Hypothesis, every productively Lindelof space is productively Hurewicz

Measure theory

Every Rothberger subset of the real line is a strong measure zero set

Function spaces

Let $X$ be a Tychonoff space, and $C(X)$ be the space of continuous functions $f\colon X\to {\mathbb {R}}$ with pointwise convergence topology

$X$ is a $\gamma$ -space iff $C(X)$ is Fréchet–Urysohn iff $C(X)$ is strong Fréchet–Urysohn
$X$ satisfies ${\text{S}}_{1}(\mathbf {\Omega } ,\mathbf {\Omega } )$ iff $C(X)$ has countable strong fan tightness
$X$ satisfies ${\text{S}}_{\text{fin}}(\mathbf {\Omega } ,\mathbf {\Omega } )$ iff $C(X)$ has countable fan tightness

Scheepers' diagram

References

↑ Menger, Karl (1924). "Einige Überdeckungssätze der punktmengenlehre". Sitzungsberichte der Wiener Akademie. 133: 421–444. doi:10.1007/978-3-7091-6110-4_14.
↑ Hurewicz, Witold (1926). "Über eine verallgemeinerung des Borelschen Theorems". Mathematische Zeitschrift. 24.1: 401–421. doi:10.1007/bf01216792.
↑ Scheepers, Marion (1996). "Combinatorics of open covers I: Ramsey theory". Topology and its Applications. 69: 31–62. doi:10.1016/0166-8641(95)00067-4.
↑ Scheepers, Marion (2001). "Selection principles in topology: New directions". Filomat. 15: 111––126.

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