Hurewicz space

For subsets of the real line, the Hurewicz property can be characterized using continuous functions into 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 ${\displaystyle n}$. A subset ${\displaystyle A}$ of ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$ is bounded if there is a function ${\displaystyle g\in \mathbb {N} ^{\mathbb {N} }}$such that ${\displaystyle f\leq ^{*}g}$ for all functions ${\displaystyle f\in A}$. A subset of ${\displaystyle \mathbb {N} ^{\mathbb {N} }}$ is unbounded if it is not bounded. Hurewicz proved that a subset of the real line is Hurewicz iff every continuous image of that space into the Baire space is unbounded. In particular, every subset of the real line of cardinality less than the bounding number ${\displaystyle {\mathfrak {b}}}$ is Hurewicz.

Let ${\displaystyle X}$ be a topological space. The Hurewicz game played on ${\displaystyle X}$ is a game with two players Alice and Bob.

1st round: Alice chooses an open cover ${\displaystyle {\mathcal {U}}_{1}}$ of ${\displaystyle X}$. Bob chooses a finite set ${\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1}}$.

2nd round: Alice chooses an open cover ${\displaystyle {\mathcal {U}}_{2}}$ of ${\displaystyle X}$. Bob chooses a finite set ${\displaystyle {\mathcal {F}}_{2}\subset {\mathcal {U}}_{2}}$.

etc.

If every point of the space ${\displaystyle X}$ belongs to all but finitely many sets ${\displaystyle \bigcup {\mathcal {F}}_{1},\bigcup {\mathcal {F}}_{2},\ldots }$ , then Bob wins the Hurewicz game. Otherwise, Alice wins.

A player has a winning strategy if he knows how to play in order to win the game (formally, a winning strategy is a function).

A topological space is Hurewicz iff Alice has no winning strategy in the Hurewicz game played on this space.[6]

A Tychonoff space ${\displaystyle X}$ is Hurewicz iff for every compact space ${\displaystyle C}$ containing the space ${\displaystyle X}$, and a ${\displaystyle G_{\delta }}$ subset G of ${\displaystyle C}$ containing the space ${\displaystyle X}$, there is a ${\displaystyle \sigma }$-compact set ${\displaystyle Y}$ with ${\displaystyle X\subset Y\subset G}$.[2]