Selection principle

In mathematics, selection principles deal with properties characterized by the possibility of obtaining mathematically significant objects by selecting 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.

Background and definitions

In 1924, Menger ^[1] introduced Menger's basis property for a metric space in which every basis of the space contains a sequence of open sets, with diameter tending to zero, such that the sequence covers the space. Soon thereafter, Hurewicz ^[2] observed that Menger's basis property can be reformulated as follows: for every sequence of open covers of the space one can choose a finite subset from every cover such that the union of the chosen sets covers the space.

Hurewicz's reformulation was the first of several important classes of topological spaces described by a selection principle from a sequence of covers. Denote by $\mathcal{O}$ the collection of all open covers of a topological space $X$, not containing $X$ itself, and let $\mathcal{A}$ and $\mathcal{B}$ be nonempty subsets of $\mathcal{O}$. Scheepers ^[3] introduced the convenient notation for the following properties:

• $\text{S}_1(\mathcal{A},\mathcal{B})$: for every sequence $\mathcal{U}_1,\mathcal{U}_2,\ldots\in\mathcal{A}$ there are $U_1\in\mathcal{U}_1,U_2\in\mathcal{U}_2,\dots$ such that $\{U_n:n\in\mathbb{N}\}\in\mathcal{B}$.
• $\text{S}_{\text{fin}}(\mathcal{A},\mathcal{B})$: for every sequence $\mathcal{U}_1,\mathcal{U}_2,\ldots\in\mathcal{A}$ there are finite $F_1\subseteq\mathcal{U}_1,F_2\subseteq\mathcal{U}_2,\dots$ such that $\bigcup_n F_n\in\mathcal{B}$.
• $\text{U}_{\text{fin}}(\mathcal{A},\mathcal{B})$: for every sequence $\mathcal{U}_1,\mathcal{U}_2,\ldots\in\mathcal{A}$, none containing a finite subcover, there are finite $F_1\subseteq\mathcal{U}_1,F_2\subseteq\mathcal{U}_2,\dots$ such that $\{\bigcup F_n:n\in\mathbb{N}\}\in\mathcal{B}$.
• $\binom{\mathcal{A}}{\mathcal{B}}$: Every member of $\mathcal{A}$ has a subset which is a member of $\mathcal{B}$.

In Sheepers' notation, Hurewicz's observation is that for metric spaces, Menger's basis property is equivalent to $\text{S}_{\text{fin}}(\mathcal{O},\mathcal{O})$. A topological space holding this property is called a Menger space. Menger spaces generalize Lindelöf spaces and compactness in a natural way and, indeed, every σ-compact space (a countable union of compact spaces) satisfies $\text{S}_{\text{fin}}(\mathcal{O},\mathcal{O})$.

A cover $\mathcal{U}\in\mathcal{O}$ of $X$ is called a $\gamma$-cover if every $x\in X$ belongs to almost all $U\in\mathcal{U}$. Denote the collection of $\gamma$-covers of $X$ by $\Gamma$. A topological space is a Hurewicz space if it hold the property $\text{U}_{\text{fin}}(\mathcal{O},\mathcal{\Gamma})$.

Examples and properties

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 $\mathbb{N}^\mathbb{N}$.

References

1. Menger, Karl (1924). "Einige Überdeckungssätze der punktmengenlehre". Sitzungsberichte der Wiener Akademie 133: 421–444.
2. Hurewicz, Witold (1926). "Über eine verallgemeinerung des Borelschen Theorems". Mathematische Zeitschrift 24.1: 401–421.
3. Scheepers, Marion (1996). "Combinatorics of open covers I: Ramsey theory". Topology and its Applications 69: 31–62.