# DBSCAN - is border point a cluster of a single point?

I have a question about popular clustering algorithm DBSCAN (https://www.aaai.org/Papers/KDD/1996/KDD96-037.pdf).

Authors of this paper propose definition of a cluster:

Let $$D$$ be a database of points. A cluster $$C$$ with respect to $$\varepsilon$$ and $$MinPts$$ is non empty subset of D satisfying the following conditions:

1. $$\forall p, q$$: if $$p \in C$$ and $$q$$ is density reachable from $$p$$, then $$q \in C$$
2. $$\forall p, q \in C$$: $$p$$ is density connected to $$q$$

Definitions of density-connectivity and density-reachability are stated in paper (definitions 3 and 4).

My question is: is a boarder point $$p$$ a cluster with one point?

It satisfies both conditions:

1. There are no points density-reachable from $$p$$, so 1. is ok.
2. $$p$$ is in eps-neighborhood of a core point (because it's a border point), therefore $$p$$ is density connected to itself.

I wouldn't have problem with that, if later in the paper authors wouldn't state that

cluster contains at least MinPts points

Do you have any ideas what might be going on here?