A note on f.p.p. and ${\displaystyle f^{\cdot }.p.p.}$

In [3], Kinoshita defined the notion of ${\displaystyle f^{\cdot }.p.p.}$ and he proved that each compact AR has ${\displaystyle f^{\cdot }.p.p.}$ In [4], Yonezawa gave some examples of not locally connected continua with f.p.p., but without ${\displaystyle f^{\cdot }.p.p.}$ In general, for each n=1,2,..., there is an n-dimensional continuum ${\displaystyle X_n}$ with f.p.p., but without ${\displaystyle f^{\cdot }.p.p.}$ such that ${\displaystyle X_n}$ is locally (n-2)-connected (see [4, Addendum]). In this note, we show that for each n-dimensional continuum X which is locally (n-1)-connected, X has f.p.p. if and only if X has ${\displaystyle f^{\cdot }.p.p.}$