Smooth continuum

at a point $p$

A continuum $X$ such that for each sequence $x_1,\ldots,x_n,\ldots,$ of points of $X$ converging towards a point $x$ and each subcontinuum $K\subset X$ containing $p$ and $x$ there exist a sequence of subcontinua $\{K_n\}$ in $X$, $p,x_n\in K_n$, converging towards $K$. A continuum that is smooth at each one of its points is called smooth.

Comments

Smoothness has a slightly different definition in the class of uniquely arcwise-connected continua, or dendroids (a continuum $X$ is uniquely arcwise-connected if for every $x$ and $y$ in $X$ there is a unique arc $[x,y]$ in $X$ connecting $x$ and $y$). One calls a uniquely arcwise-connected continuum $X$ smooth if there is a $p\in X$ such that $[p,x_n]\to[p,x]$ whenever $x_n\to x$ in $X$. Such a point $p$ is called an initial point of $X$.

References

[a1]	J.J. Charatonik, C. Eberhart, "On smooth dendroids" Fund. Math. , 67 (1970) pp. 297–322