A betweenness structure on a set X is a ternary relation $[\cdot ,\cdot ,\cdot ]\subseteq X^3$ that captures a rudimentary notion of one point of X lying between two others. The interval $[a,b]$ is the set of all points lying between a and b, and a subset C of X is convex if $[a,b]\subseteq C$ whenever $a,b\in C$. The span of a set A is the union of all intervals $[a,b]$, where $a,b\in A$; by iterating the span operator countably many times, we obtain the convex hull of A. The betweenness structure is topological if X carries a topology that satisfies certain compatibility conditions with respect to betweenness; in particular, intervals are closed subsets. We are guided by questions involving how the span and convex hull operators interact with the topological closure and interior operators, especially in the domains of metric spaces and of continua. With a metric space $〈X,\varrho 〉$, $[a,c,b]$ holds exactly when $\varrho (a,b)=\varrho (a,c)+\varrho (c,b)$; and one result about this betweenness structure is that the span of any compact subset is both closed and bounded. With a continuum X, $[a,c,b]$ holds exactly when c belongs to every subcontinuum of X that contains a and b; and one result about this betweenness structure is that when the continuum is either aposyndetic or hereditarily unicoherent, the closure of a convex subset is always convex.

甲介结构上的一组X是三元关系$[\cdot ,\cdot ,\cdot ]\subseteq X^3$这捕获了X的一个点位于其他两个点之间的基本概念。该区间 $[\mathrm{一种},乙]$是集之间躺在所有点的一个和b，以及一个子集ç的X是凸如果$[\mathrm{一种},乙]\subseteq C$ 每当 $\mathrm{一种},乙\in C$. 集合A的跨度是所有区间的并集$[\mathrm{一种},乙]$， 在哪里 $\mathrm{一种},乙\in \mathrm{一种}$; 通过迭代操作者跨度可数很多次，我们得到的凸包的甲。如果X携带的拓扑满足关于介数的某些兼容性条件，则介数结构是拓扑的；特别是，区间是封闭的子集。我们以涉及跨度和凸包算子如何与拓扑闭包和内部算子相互作用的问题为指导，特别是在度量空间和连续体领域。有度量空间$〈X,ㄧ〉$, $[\mathrm{一种},C,乙]$ 恰好在什么时候成立 $ㄧ(\mathrm{一种},乙)=ㄧ(\mathrm{一种},C)+ㄧ(C,乙)$; 关于这种介数结构的一个结果是任何紧凑子集的跨度都是封闭的和有界的。有一个连续的X，$[\mathrm{一种},C,乙]$当c属于包含a和b的X 的每个子连续统时，恰好成立；并且关于这种介数结构的一个结果是，当连续统是离合的或遗传单相干的时，凸子集的闭包总是凸的。