Towards the Complete Classification of the Two Lines Inverse Limit

Abstract

AbstractFor positive numbers $$r<1$$ r < 1 and $$\rho >1$$ ρ > 1 , let $$L_{r,\rho }$$ L r , ρ be the union of two line segments in $$ [0,1] \times {[}0,1{]}$$ [ 0 , 1 ] × [ 0 , 1 ] , one from (0, 0) to (1, r) and the other one from (0, 0) to $$(\frac{1}{\rho },1)$$ ( 1 ρ , 1 ) . In Banič et al. (2022), it was proven for all such r and $$\rho $$ ρ that, if r and $$\rho $$ ρ never connect, then the Mahavier product of $$L_{r,\rho }$$ L r , ρ ’s is homeomorphic to the Lelek fan. In this paper, we show that for all such r and $$\rho $$ ρ , if r and $$\rho $$ ρ do connect, the Mahavier product of $$L_{r,\rho }$$ L r , ρ ’s is a fan F with top v with some additional properties. More precisely, it is the union of a countable family $$\mathcal {C}$$ C of Cantor fans such that for each $$C_1,C_2\in \mathcal {C}$$ C 1 , C 2 ∈ C , if $$C_1\ne C_2$$ C 1 ≠ C 2 , then $$C_1\cap C_2 = \{v\}$$ C 1 ∩ C 2 = { v } and the set of limit points of the set of end-points of F, forms in each arc from v to an end-point, a harmonic sequence. This solves the open problem from Banič et al. (2022).