On the hyperspaces of meager and regular continua

Given a metric continuum X, we consider the collection of all regular subcontinua of X and the collection of all meager subcontinua of X, these hyperspaces are denoted by D (X) and M (X), respectively. It is known that D (X) is compact if and only if D (X) is finite. In this way, we find some conditions related about the cardinality of D (X) and we reduce the fact to count the elements of D (X) to a Graph Theory problem, as an application of this, we prove in particular that | D (X) | ∉ 2, 3, 4, 5, 8, 9 h t) for any continuum X. Also, we prove that D (X) is never homeomorphic to ℕ. On the other hand, given a point p ∈ X, we consider the meager composant and the filament composant of p in X, denoted by MXp and FcsX (p), respectively, and we study some relations between MXp and FcsX (p) such as the equality of them as a subset of X. Also, we construct examples showing that the collection Fcs (X) = FcsX (p): p ∈ X can be homeomorphic to: any finite discrete space, the harmonic sequence, the closure of the harmonic sequence and the Cantor set. Finally, we study the contractibility of M (X) ; we prove the arc of pseudo-arcs, which is a no contractible continuum, satisfies that its hyperspace of meager subcontinua is contractible, given a solution to an open problem. Also, we rise open problems.