The Second Main Theorem: Exceptional Branch Points; The Condition k>l

2012 ◽  
pp. 69-97
Author(s):  
Anthony Tromba
Keyword(s):  
Branch Points ◽  
Second Main Theorem ◽  
The Second Main Theorem

Proof of the Second Main Theorem

2017 ◽  
pp. 149-191
Author(s):  
Marian B. Pour-El ◽  
J. Ian Richards
Keyword(s):  
Second Main Theorem ◽  
The Second Main Theorem

scholarly journals PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES

2017 ◽  
Vol 82 (4) ◽  
pp. 1317-1355
Author(s):  
PHILIPP SCHLICHT
Keyword(s):  
Baire Space ◽  
Inaccessible Cardinal ◽  
Second Main Theorem ◽  
Perfect Set ◽  
Uncountable Cardinal ◽  
First Main Theorem ◽  
Definable Functions ◽  
The Second Main Theorem

AbstractWe extend Solovay’s theorem about definable subsets of the Baire space to the generalized Baire spaceλλ, whereλis an uncountable cardinal withλ<λ= λ. In the first main theorem, we show that the perfect set property for all subsets ofλλthat are definable from elements ofλOrd is consistent relative to the existence of an inaccessible cardinal aboveλ. In the second main theorem, we introduce a Banach–Mazur type game of lengthλand show that the determinacy of this game, for all subsets ofλλthat are definable from elements ofλOrd as winning conditions, is consistent relative to the existence of an inaccessible cardinal aboveλ. We further obtain some related results about definable functions onλλand consequences of resurrection axioms for definable subsets ofλλ.

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