TITLE: What is a 0-ball?
QUESTION [6 upvotes]: I'm reading a paper that says $\bigcap V_{T,X}$ is either empty or a closed $l$-ball where $T \subset S$ is a subset of points $S$ and $\operatorname{card}{T} = m + 1 - l$ where $m$ is the dimension of the smooth manifold $\Sigma$ that the points $S$ are sampled from. My question is what happens when $l=0$, meaning that $\operatorname{card}{T} = m+1$. In this case the intersection must be a closed $0$-ball. How is a $0$-ball defined and what does it mean for an intersection of two topological spaces to be a $0$-ball?

REPLY [5 votes]: A zero-ball is a point.
Interestingly, however, the zero-sphere consists of two points! Indeed, the zero-sphere is the boundary of the $1$-ball, which is an interval...