A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every closed set contains a closed point. However, neither of us are topologists, so we don't know whether this exists in the literature yet. Does it? If so, what is it called, and where can I find information on it? If not, I'm happy to just call such a space "pearled" (intuition: the closed sets are the oysters), but I thought I'd ask here before publishing an existing definition under a new name.

By the way, every T$_1$-space is pearled, as is every finite T$_0$-space and every spectral space, but the property of being pearled is independent of the T$_0$-property. However, I would be interested even in a name for a pearled T$_0$-space. Is this the same as a T$_0$-space whose lattice of closed sets is atomic?

A coauthor and I have stumbled upon a useful topological property -- namely, we are interested in the property that every nonempty closed set contains a closed point. However, neither of us are topologists, so we don't know whether this exists in the literature yet. Does it? If so, what is it called, and where can I find information on it? If not, I'm happy to just call such a space "pearled" (intuition: the closed sets are the oysters), but I thought I'd ask here before publishing an existing definition under a new name.

By the way, every T$_1$-space is pearled, as is every finite T$_0$-space and every spectral space, but the property of being pearled is independent of the T$_0$-property. However, I would be interested even in a name for a pearled T$_0$-space. Is this the same as a T$_0$-space whose lattice of closed sets is atomic?