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Diffstat (limited to 'sections/preliminaries.tex')
| -rw-r--r-- | sections/preliminaries.tex | 6 |
1 files changed, 3 insertions, 3 deletions
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex index 266845c..82e64b4 100644 --- a/sections/preliminaries.tex +++ b/sections/preliminaries.tex @@ -85,7 +85,7 @@ $\bigcap_{n}V_n \subseteq A$. \end{definition} - There is an important theorem \ref{theorem:banach_mazur_thm} on the + There is an important Theorem \ref{theorem:banach_mazur_thm} on the Banach-Mazur game: $A$ is comeagre if and only if $\textit{II}$ can always choose sets $V_0, V_1, \ldots$ such that it wins. Before we prove it we need to define notions necessary to formalise and prove the theorem. @@ -254,7 +254,7 @@ Now we can move to the proof of the Banach-Mazur theorem. - \begin{proof}[Proof of theorem \ref{theorem:banach_mazur_thm}] + \begin{proof}[Proof of Theorem \ref{theorem:banach_mazur_thm}] $\Rightarrow$: Let $(A_n)$ be a sequence of dense open sets with $\bigcap_n A_n\subseteq A$. The simply $\textit{II}$ plays $V_n = U_n\cap A_n$, which is nonempty by the denseness of $A_n$. @@ -277,7 +277,7 @@ \begin{corollary} \label{corollary:banach-mazur-basis} If we add a constraint to the Banach-Mazur game such that players can only - choose basic open sets, then the theorem \ref{theorem:banach_mazur_thm} + choose basic open sets, then the Theorem \ref{theorem:banach_mazur_thm} still suffices. \end{corollary} |