Yes, that's one of the things that makes it a cone. Just like a pyramid has a vertex as well.
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No, a cone has one vertex.
A cone only has 1 vertex because it only has one point where it connects. (vertex=a point)
You could call the vertex, the apex, the tip, or the point of the cone.
A cone has one vertex at the tip.
A vertex is a point, so one.