Suggestions for Reverse Mathematics beyond $L_{2}$

BlogSuggestions for Reverse Mathematics beyond $L_{2}$
Sam Sanders asked 3 years ago

The most common framework for classical Reverse Mathematics is no doubt second-order arithmetic.  Kohlenbach has formulated a higher-order framework (See his RM2001 paper), and recent results by Normann-Sanders (arxiv link) suggest that very basic higher-order theorems (like ‘the unit interval is open-cover compact’) fall far outside of the Big Five, and do not even fit the linear order of the Godel hierarchy.      
I am soliciting suggestions for similarly promising (but as basic as possible) topics to be studied in the higher-order framework.  
To give the reader an idea: the following topics have been studied by Normann-Sanders, and behave as mentioned above (i.e. fall outside of the Godel hierarchy).  

  1. open cover (aka Heine-Borel) compactness of the unit interval (including uncountable covers).
  2. Other covering lemmas, including Vitali’s, and related properties, like Lebesgue numbers (including uncountable covers). 
  3. The Lindeloef property of $\mathbb{R}$.
  4. paracompactness of the unit interval (including uncountable covers).
  5. basic properties of the gauge integral (uniqueness, extension of Lebesgue and Riemann integrals,..).
  6. basic properties of the Lebesgue integral, formulated as a restricted gauge integral.
  7. basic properties of the Lebesgue integral, formulated with Kreuzer’s $\lambda^3$.
  8. Egorov and Lusin theorem for both formulations of the Lebesgue integral. 
  9. The neighbourhood function principle (NFP) from intuitionistic mathematics. 
  10. Urysohn’s identity for the unit interval.
  11. Uniform statements like: “A continuous function F on [0,1] has a modulus of uniform continuity AND the latter only depends on a modulus of continuity of F”.