Selection principle

In 1924, Karl Menger [1] introduced the following basis property for metric spaces: Every basis of the topology contains a sequence of sets with vanishing diameters that covers the space. Soon thereafter, Witold Hurewicz[2] observed that Menger's basis property is equivalent to the following selective property: for every sequence of open covers of the space, one can select finitely many open sets from each cover in the sequence, such that the selected sets cover the space. Topological spaces having this covering property are called Menger spaces.

Hurewicz's reformulation of Menger's property was the first important topological property described by a selection principle. Let ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {B} }$ be classes of mathematical objects. In 1996, Marion Scheepers[3] introduced the following selection hypotheses, capturing a large number of classic mathematical properties:

• ${\displaystyle {\text{S}}_{1}(\mathbf {A} ,\mathbf {B} )}$: For every sequence ${\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots }$ of elements from the class ${\displaystyle \mathbf {A} }$, there are elements ${\displaystyle U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\dots }$ such that ${\displaystyle \{U_{n}:n\in \mathbb {N} \}\in \mathbf {B} }$.
• ${\displaystyle {\text{S}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )}$: For every sequence ${\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots }$ of elements from the class ${\displaystyle \mathbf {A} }$, there are finite subsets ${\displaystyle {\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots }$ such that ${\displaystyle \bigcup _{n=1}^{\infty }{\mathcal {F}}_{n}\in \mathbf {B} }$.

In the case where the classes ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {B} }$ consist of covers of some ambient space, Scheepers also introduced the following selection principle.

• ${\displaystyle {\text{U}}_{\text{fin}}(\mathbf {A} ,\mathbf {B} )}$: For every sequence ${\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots }$ of elements from the class ${\displaystyle \mathbf {A} }$, none containing a finite subcover, there are finite subsets ${\displaystyle {\mathcal {F}}_{1}\subseteq {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subseteq {\mathcal {U}}_{2},\dots }$ such that ${\displaystyle \{\bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\dotsc \}\in \mathbf {B} }$.

Later, Boaz Tsaban identified the prevalence of the following related principle:

• ${\displaystyle {\binom {\mathbf {A} }{\mathbf {B} }}}$: Every member of the class ${\displaystyle \mathbf {A} }$ contains a member of the class ${\displaystyle \mathbf {B} }$.

The notions thus defined are selection principles. An instantiation of a selection principle, by considering specific classes ${\displaystyle \mathbf {A} }$ and ${\displaystyle \mathbf {B} }$, gives a selection (or: selective) property. However, these terminologies are used interchangeably in the literature.

Covering properties form the kernel of the theory of selection principles. Selection properties that are not covering properties are often studied by using implications to and from selective covering properties of related spaces.

Let ${\displaystyle X}$ be a topological space. An open cover of ${\displaystyle X}$ is a family of open sets whose union is the entire space ${\displaystyle X.}$ For technical reasons, we also request that the entire space ${\displaystyle X}$ is not a member of the cover. The class of open covers of the space ${\displaystyle X}$ is denoted by ${\displaystyle \mathbf {O} }$. (Formally, ${\displaystyle \mathbf {O} (X)}$, but usually the space ${\displaystyle X}$ is fixed in the background.) The above-mentioned property of Menger is, thus, ${\displaystyle {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )}$. In 1942, Fritz Rothberger considered Borel's strong measure zero sets, and introduced a topological variation later called Rothberger space (also known as C${\displaystyle ''}$ space). In the notation of selections, Rothberger's property is the property ${\displaystyle {\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )}$.

An open cover ${\displaystyle {\mathcal {U}}}$ of ${\displaystyle X}$ is point-cofinite if it has infinitely many elements, and every point ${\displaystyle x\in X}$ belongs to all but finitely many sets ${\displaystyle U\in {\mathcal {U}}}$. (This type of cover was considered by Gerlits and Nagy, in the third item of a certain list in their paper. The list was enumerated by Greek letters, and thus these covers are often called ${\displaystyle \gamma }$-covers.) The class of point-cofinite open covers of ${\displaystyle X}$ is denoted by ${\displaystyle \mathbf {\Gamma } }$. A topological space is a Hurewicz space if it satisfies ${\displaystyle {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )}$.

An open cover ${\displaystyle {\mathcal {U}}}$ of ${\displaystyle X}$ is an ${\displaystyle \omega }$-cover if every finite subset of ${\displaystyle X}$ is contained in some member of ${\displaystyle {\mathcal {U}}}$. The class of ${\displaystyle \omega }$-covers of ${\displaystyle X}$ is denoted by ${\displaystyle \mathbf {\Omega } }$. A topological space is a γ-space if it satisfies ${\displaystyle {\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}}$.

By using star selection hypotheses one obtains properties such as star-Menger (${\displaystyle {\text{S}}_{\text{fin}}^{*}(\mathbf {O} ,\mathbf {O} )}$), star-Rothberger (${\displaystyle {\text{S}}_{1}^{*}(\mathbf {O} ,\mathbf {O} )}$) and star-Hurewicz (${\displaystyle {\text{S}}_{\text{fin}}^{*}(\mathbf {O} ,\mathbf {\Gamma } )}$).

The Scheepers Diagram

There are 36 selection properties of the form ${\displaystyle \Pi (\mathbf {A} ,\mathbf {B} )}$, for ${\displaystyle \Pi \in \{{\text{S}}_{1},{\text{S}}_{\text{fin}},{\text{U}}_{\text{fin}},{\bigl (}~~{\bigr )}\}}$ and ${\displaystyle \mathbf {A} ,\mathbf {B} \in \{\mathbf {O} ,\mathbf {\Gamma } ,\mathbf {\Omega } \}}$. Some of them are trivial (hold for all spaces, or fail for all spaces). Restricting attention to Lindelöf spaces, the diagram below, known as the Scheepers Diagram,[3][5] presents nontrivial selection properties of the above form, and every nontrivial selection property is equivalent to one in the diagram. Arrows denote implications.

Selection principles also capture important non-covering properties.

Let ${\displaystyle Y}$ be a topological space, and ${\displaystyle y\in Y}$. The class of sets ${\displaystyle A}$ in the space ${\displaystyle Y}$ that have the point ${\displaystyle y}$ in their closure is denoted by ${\displaystyle \mathbf {\Omega _{y}} }$. The class ${\displaystyle \mathbf {\Omega _{y}^{\text{ctbl}}} }$ consists of the countable elements of the class ${\displaystyle \mathbf {\Omega _{y}} }$. The class of sequences in ${\displaystyle Y}$ that converge to ${\displaystyle y}$ is denoted by ${\displaystyle \mathbf {\Gamma _{y}} }$.

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

There are close connections between selection principles and Topological Games.

• Every ${\displaystyle {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )}$ space is a Lindelöf space.
• Every σ-compact space (a countable union of compact spaces) is ${\displaystyle {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )}$.
• ${\displaystyle {\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}\Rightarrow {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )\Rightarrow {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )}$.
• ${\displaystyle {\binom {\mathbf {\Omega } }{\mathbf {\Gamma } }}\Rightarrow {\text{S}}_{1}(\mathbf {O} ,\mathbf {O} )\Rightarrow {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )}$.
• Assuming the Continuum Hypothesis, there are sets of real numbers witnessing that the above implications cannot be reversed.[5]
• Every Luzin set is ${\displaystyle {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )}$ but no ${\displaystyle {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )}$.[11][12]
• Every Sierpiński set is Hurewicz.[13]

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

• A subset of the real line is ${\displaystyle {\text{S}}_{\text{fin}}(\mathbf {O} ,\mathbf {O} )}$ if and only if every continuous image of that space into the Baire space is not dominating.[14]
• A subset of the real line is ${\displaystyle {\text{U}}_{\text{fin}}(\mathbf {O} ,\mathbf {\Gamma } )}$ if and only if every continuous image of that space into the Baire space is bounded.[14]

• Compact space
• Sigma-compact
• Menger space
• Hurewicz space
• Rothberger space