<p>Endowing ordinals with the order topology, a complete description of the Wadge hierarchy restricted to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$B(\omega _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> —the family of Borel subsets of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\omega _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ω</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>— is given. Using this, it is shown that if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\alpha <\omega _1^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo><</mo> <msubsup> <mi>ω</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mrow> </math></EquationSource> </InlineEquation> the Wadge hierarchy on the Borel subsets of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is a bqo. Also, for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\alpha \ge \omega _{\omega }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <msub> <mi>ω</mi> <mi>ω</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> the Wadge hierarchy on the Borel subsets of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> has infinite antichains, so it is not a wqo. This leaves the question open for <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\omega _1^2\le \alpha <\omega _\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>ω</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mo>≤</mo> <mi>α</mi> <mo><</mo> <msub> <mi>ω</mi> <mi>ω</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>; however it is shown that the method used for smaller ordinals cannot be employed in this case. The results for the order topology also allow to obtain the Wadge hierarchy restricted to <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$B(\omega _1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for the compact complement topology.</p>

The Wadge hierarchy on $\omega _1$ and some of its consequences

期刊论文
发表日期：

2026年3月13日

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Riccardo Camerlo

摘要

Endowing ordinals with the order topology, a complete description of the Wadge hierarchy restricted to $B(\omega _1)$ $B (ω_{1})$ —the family of Borel subsets of $\omega _1$ $ω_{1}$ — is given. Using this, it is shown that if $\alpha <\omega _1^2$ $α < ω_{1}^{2}$ the Wadge hierarchy on the Borel subsets of $\alpha $ $α$ is a bqo. Also, for $\alpha \ge \omega _{\omega }$ $α \geq ω_{ω}$ the Wadge hierarchy on the Borel subsets of $\alpha $ $α$ has infinite antichains, so it is not a wqo. This leaves the question open for $\omega _1^2\le \alpha <\omega _\omega $ $ω_{1}^{2} \leq α < ω_{ω}$ ; however it is shown that the method used for smaller ordinals cannot be employed in this case. The results for the order topology also allow to obtain the Wadge hierarchy restricted to $B(\omega _1)$ $B (ω_{1})$ for the compact complement topology.

The Wadge hierarchy on \(\omega _1\) and some of its consequences

摘要