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❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u < d ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❑✉rt ●ö❞❡❧ ❘❡s❡❛r❝❤ ❈❡♥t❡r ❢♦r ▼❛t❤❡♠❛t✐❝❛❧ ▲♦❣✐❝✱ ❯♥✐✈❡rs✐t② ♦❢ ❱✐❡♥♥❛ ▲♦❣✐❝ ❈♦❧❧♦q✉✐✉♠ ✷✵✵✼ ❲r♦❝➟❛✇ ❏✉❧② ✶✸ ✕ ✷✵✱ ✷✵✵✼ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ❖✉t❧✐♥❡ ✶ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ✷ ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ✸ ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ✹ ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❢ ❝♦♥t❛✐♥s ❧❡ss ✐♥❢♦r♠❛t✐♦♥ t❤❛♥ ✿ ❢ ❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ▼❛♣♣✐♥❣s ❜❡t✇❡❡♥ ✜❧t❡rs ❉❡✜♥✐t✐♦♥ ❆ ✜❧t❡r ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♣r♦♣❡r ✜❧t❡r ♦♥ ω ✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❢ : ω → ω ❜❡ ✜♥✐t❡✲t♦✲♦♥❡✳ ❲❡ s❡t ❢ ( F ) = { ❳ : ❢ − ✶ ❳ ∈ F } ✳ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ▼❛♣♣✐♥❣s ❜❡t✇❡❡♥ ✜❧t❡rs ❉❡✜♥✐t✐♦♥ ❆ ✜❧t❡r ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♣r♦♣❡r ✜❧t❡r ♦♥ ω ✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❢ : ω → ω ❜❡ ✜♥✐t❡✲t♦✲♦♥❡✳ ❲❡ s❡t ❢ ( F ) = { ❳ : ❢ − ✶ ❳ ∈ F } ✳ ❢ ( F ) ❝♦♥t❛✐♥s ❧❡ss ✐♥❢♦r♠❛t✐♦♥ t❤❛♥ F ✿ ✲ F ✘ ✘ ✘ � ✘ ✘ ✘ ✘ � ✘ ✾ � ✠ ✲ ❢ ( F ) ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ▼❛♣♣✐♥❣s ❜❡t✇❡❡♥ ✜❧t❡rs ❉❡✜♥✐t✐♦♥ ❆ ✜❧t❡r ✐s ❛ ♥♦♥✲♣r✐♥❝✐♣❛❧ ♣r♦♣❡r ✜❧t❡r ♦♥ ω ✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❢ : ω → ω ❜❡ ✜♥✐t❡✲t♦✲♦♥❡✳ ❲❡ s❡t ❢ ( F ) = { ❳ : ❢ − ✶ ❳ ∈ F } ✳ ❢ ( F ) ❝♦♥t❛✐♥s ❧❡ss ✐♥❢♦r♠❛t✐♦♥ t❤❛♥ F ✿ ✲ F ✘ ✘ ✘ � ✘ ✘ ✘ ✘ � ✘ ✾ � ✠ ✲ ❢ ( F ) ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❚✇♦ ✉❧tr❛✜❧t❡rs ❛♥❞ ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ s✉❝❤ t❤❛t ❢ ❢ ✳ ❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ ( F ) ∪ ❢ ( G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ ( U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ ( F ) ∪ ❢ ( G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ ( U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❚✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ ( U ) = ❢ ( V ) ✳ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u

❖✉t❧✐♥❡ ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❚❤❡ ❝❛s❡ ♦❢ ✉❧tr❛✜❧t❡rs ❊①♣❧❛♥❛t✐♦♥ ♦❢ t❤❡ ❝❛r❞✐♥❛❧ ❝❤❛r❛❝t❡r✐st✐❝s ❆ ♠♦❞❡❧ ♦❢ u < d ✇✐t❤ ✐♥✜♥✐t❡❧② ♠❛♥② ❝❧❛ss❡s ◆❡❛r ❝♦❤❡r❡♥❝❡ ♦❢ ✜❧t❡rs ❉❡✜♥✐t✐♦♥ ❚✇♦ ✜❧t❡rs F ❛♥❞ G ♦♥ ω ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ ( F ) ∪ ❢ ( G ) ❣❡♥❡r❛t❡s ❛ ♣r♦♣❡r ✜❧t❡r✳ ■❢ U ✐s ❛♥ ✉❧tr❛✜❧t❡r✱ t❤❡♥ ❛❧s♦ ❢ ( U ) ✐s ❛♥ ✉❧tr❛✜❧t❡r✳ ❚✇♦ ✉❧tr❛✜❧t❡rs U ❛♥❞ V ❛r❡ ♥❡❛r❧② ❝♦❤❡r❡♥t ✐❢ t❤❡r❡ ✐s ❛ ✜♥✐t❡✲t♦✲♦♥❡ ❢✉♥❝t✐♦♥ ❢ : ω → ω s✉❝❤ t❤❛t ❢ ( U ) = ❢ ( V ) ✳ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❚❤❡r❡ ♠❛② ❜❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♥❡❛r ❝♦❤❡r❡♥❝❡ ❝❧❛ss❡s ✉♥❞❡r u