Inductively approaching subsets by almost finite sets, we refute Scheepers’ conjecture under CH. More precisely, we prove the following. 

　　1. Assuming CH, there is a subset of reals $X$ such that $C_p(X)$ has property ($\alpha _2$) and $X$ does not satisfy $S_1(\Gamma , \Gamma )$. 

　　Applying the idea of approaching subsets by almost finite sets and using an analogous approaching, we complete the Scheepers Diagram. 

　　2. $U_{fin}(\Gamma , \Gamma )$ implies $S_{fin}(\Gamma , \Omega )$. 

　　3. $U_{fin}(\Gamma , \Omega )$ does not imply $S_{fin}(\Gamma , \Omega )$. More precisely, assuming CH, there is a subset of reals $X$ satisfying $U_{fin}(\Gamma , \Omega )$ such that $X$ does not satisfy $S_{fin}(\Gamma , \Omega )$. 

　　These results solve three longstanding and major problems in selection principles. 

　　Publication: 

　　Transactions of The American Mathematical Society, Volume 376, Number 2, February 2023, Pages 1199–1229. 

　　https://doi.org/10.1090/tran/8787 

　　Author: 

　　Yinhe Peng 

　　Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China 

　　Email: pengyinhe@amss.ac.cn