some fix for some problem
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@ -17,13 +17,13 @@
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\]
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and similarly for $<^\ast$.
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% https://q.uiver.app/#q=WzAsNCxbMCwwLCJZIl0sWzIsMCwiWCJdLFswLDEsIkMnIl0sWzIsMSwiXFxiZWdpbnthcnJheX17Y31DXFxcXFxcUGlfMV4xLVxcdGV4dHtjb21wbGV0ZX1cXGVuZHthcnJheX0iXSxbMCwxLCJmIl0sWzIsMCwiXFxzdWJzZXRlcSIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMSwiXFxzdWJzZXRlcSIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV1d
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\begin{tikzcd}
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Y && X \\
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{C'} && {\begin{array}{c}C\\\Pi_1^1-\text{complete}\end{array}}
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\arrow["f", from=1-1, to=1-3]
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\arrow["\subseteq", hook, from=2-1, to=1-1]
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\arrow["\subseteq"', hook, from=2-3, to=1-3]
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\end{tikzcd}
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% \begin{tikzcd}
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% Y && X \\
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% {C'} && {\begin{array}{c}C\\\Pi_1^1-\text{complete}\end{array}}
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% \arrow["f", from=1-1, to=1-3]
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% \arrow["\subseteq", hook, from=2-1, to=1-1]
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% \arrow["\subseteq"', hook, from=2-3, to=1-3]
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% \end{tikzcd}
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Let $X = 2^{\Q} \supseteq \WO$.
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We have already show that $\WO$ is $\Pi^1_1$-complete.
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