Given a metric continuum X, let $C(X)$ and $F_1(X)$ be the hyperspaces of subcontinua and the one-point sets of X, respectively. Let $D(X)$ be the collection of all regular subsets in X belonging to $C(X)$ and let $M(X)$ be all the subcontinua of X with empty interior. In the first part of this paper we are interested in the problem to classify continua that satisfies one of the following equalities: $F_1(X)=M(X)$, $D(X)=D(X)-F_1(X)$, $D(X)=C(X)-M(X)$ and $C(X)=D(X)\cup M(X)$. We show that for hereditarily locally connected continua these equalities and the property of not contain a continuum called dendrite ${\mathcal{D}}_1$ are equivalent each other. In the second part, we consider a continuum X for which there exists a continuous surjective and monotone function $f:X\to [0,1]$. We show that the conditions that X is a λ-type continuum and every t in $[0,1]$ is a cohesion point are equivalent to the equalities $D(X)=\{A\in C(X):f(A)\in D([0,1])\}$ and $M(X)=\{A\in C(X):f(A)\in M([0,1])\}$. Throughout this paper we pose open problems.

给定一个度量连续统X，让$C(X)$ 和 $F_1(X)$分别是次连续的超空间和X的单点集。让$D(X)$是X 中属于的所有正则子集的集合$C(X)$ 然后让 $米(X)$是X 的所有子连续体，内部为空。在本文的第一部分中，我们对满足以下等式之一的连续体进行分类的问题感兴趣：$F_1(X)=米(X)$, $D(X)=D(X)-F_1(X)$, $D(X)=C(X)-米(X)$ 和 $C(X)=D(X)\cup 米(X)$. 我们表明，对于遗传局部连接的连续体，这些等式和不包含称为树突的连续体的性质${\mathcal{D}}_1$彼此等价。在第二部分，我们考虑存在连续满射单调函数的连续统X$F：X\to [0,1]$. 我们证明了X是λ型连续体的条件，并且每个t在$[0,1]$ 是一个内聚点 等价于等式 $D(X)=\{\mathrm{一种}\in C(X)：F(\mathrm{一种})\in D([0,1])\}$ 和 $米(X)=\{\mathrm{一种}\in C(X)：F(\mathrm{一种})\in 米([0,1])\}$. 在整篇论文中，我们提出了开放性问题。