Nonblockers in homogeneous continua

Given a continuum X, an element A∈2X∖{X} is said to be a set that does not block singletons of X provided that for every point x∈X∖A, there exists an order arc α:[0,1]→C(X) such that α(0)={x}, α(1)=X and α(t)∩A=∅ for all t∈[0,1). If A does not block singletons of X, we say that A is a nonblocker of X. In this paper, we present some properties about the sets that do not block singletons and, for a homogeneous curve X, we use the set NB(F1(X)) to construct a continuous decomposition of X that is a homogeneous curve and a Whitney level of C(X), where its elements are homogeneous, acyclic, hereditarily unicoherent and indecomposable continuum. Also, we prove that if X is a continuum such that NB(F1(X))=F1(X), then the next three properties are equivalent: 1) X is a homogeneous continuum, 2) X has the property that for all x∈X, there exists a proper subcontinuum A of X such that x∈Int(A), and 3) X≈S1.