In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed to include zero.

1. Is there a standard terminology for operads with and without $O(0)$?

2. Is there standard terminology for topological operads where $O(0)$ is a point, vs. $O(0)$ not being a point?

Although it's less important I'd be curious if people have examples where these distinctions are interesting.

In the Markl, Schneider and Stasheff text, topological operads are an indexed collection of spaces $O(n)$ for $n \in \{1,2,3,\cdots\}$ satisfying some axioms. In May's text, the index set is allowed to include zero.

1. Is there a standard terminology for operads with and without $O(0)$?

2. Is there standard terminology for topological operads where $O(0)$ is a point, vs. $O(0)$ not being a point?

Although it's less important I'd be curious if people have examples where these distinctions are interesting.

Since any operad acts on its $O(0)$ part perhaps the $O(0)$ part should be called something like its "base"? But then "baseless operad" would sound kind of pejorative.