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T000141 |
| if |
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| then |
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| refs |
| doi |
name |
10.1016/S0166-8641(96)00075-2 |
The combinatorics of open covers II |
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See the proof of Theorem 2.2 in {{doi:10.1016/S0166-8641(96)00075-2}}, which we summarize here.
An $\omega$-cover of a compact space admits a finite subcover that covers all $k$-sized subsets for any positive integer $k$. Write the space $X$ as an ascending union of ${K_n:n\in\omega}$ where each $K_n$ is compact. Given an $\omega$-cover $\mathscr U_n$ of $X$, let $\mathscr V_n$ be a finite subset of $\mathscr U_n$ that covers all $n$-sized subsets of $K_n$. Note that this is a winning Markov strategy in the $\omega$-Menger game (equivalent to {P70}).