Half-homogeneous chainable continua with end points

Degree of homogeneity on cones

“…In the present paper we determine the orbits of the cone of X-in terms of those of X-for some spaces X, namely local dendrites and Hausdorff spaces with no arcs; in particular we show that no local dendrite has 1 3 -homogeneous cone. We also analyze how some subsets of Cone(X) behave under homeomorphisms of Cone(X) onto itself.…”

“…. , ω} [3, Proposition 4.7, p. 174], hence its cone is 1 7 -homogeneous. This shows that finite graphs are not the only local dendrites whose cone has a finite degree of homogeneity.…”

“…An orbit of X is the action of H(X) at a point x 0 of X, namely O X (x 0 ) = {h(x 0 ) : h ∈ H(X)}. Given a positive integer n, a space is said to be 1 n -homogeneous provided that X has exactly n orbits, in which case we say that the degree of homogeneity of X is n. Since 2006 there has been increasing interest in the study of 1 2 -homogeneity, in fact, several papers have been written on the subject: [1,2,6,[10][11][12][13][16][17][18][19][20][21][22][23][24]. Higher degrees of homogeneity appear to be studied only in [8,26].…”

“…Then: (i) d H (Cone(X)) = 2 and X is an arc or a simple closed curve or (ii) d H (Cone(X)) = 4 and X ∈ {T n : n ∈ N \ {1, 2}} ∪ {F ω } ∪ {θ m : m ∈ N \ {1, 2}}.Corollary 7.0.25. If X is a local dendrite, then Cone(X) is not1 3 -homogeneous.…”

Non-cut, shore and non-block points in continua

“…The converse need not be true. For example the two end points of the arcless arc ( [BPV13]) are absolute end points but these are not of order one. This suggests to use the following notion.…”