Some results on 1/n-homogeneity

Given a positive integer $n,$ a non-empty topological space is said to be $\frac{1}{n}$-homogeneous provided there are exactly $n$ orbits for the action of the group of homeomorphisms of the space onto itself. Now, for a non-empty topological space $X,$ the cone of $X$, $ \mathop{\rm Cone}(X),$ is the quotient space that is obtained by identifying all the points $(x,1)$ in $X\times [0,1]$ to a single point. The suspension of $X$, $\mathop{\rm Sus}(X),$ is the quotient space that is obtained by identifying all the points $(x,1)$ in $X\times [-1,1]$ to a single point, and all the points $(x,-1)$ to another point. The quotient space $Z_{X}$, is the space that is obtained by identifying all the points $(x,1)$ and all the points $(x,-1)$ in $X\times [-1,1]$ to one point. In this paper we determine general properties of the quotient spaces $Z_{X}$ and we investigate $\frac{1}{n}$-homogeneity on the quotient spaces $Z_{X}$, $\mathop{\rm Sus}(Z_X)$ and $\mathop{\rm Cone}(Z_X)$, among certain classes of compact metric spaces. In particular, we obtain the degree of homogeneity of the harmonic sequence, the $n$-rose finite graph and the space $X$ which is the union of a sequence $\{C_n\}^{\infty}_{n=1}$ of circles in the plane joined by a point and converging to a limit circle $C_0$.