What can we call a sphere that is missing a spherical sector or any other connected piece ? What can we call a sphere minus an open simple curve ?
I have found in page 96 in "Heather A. Dye,An Invitation to Knot Theory: Virtual and Classical, CRC Press, 2016" that a punctured sphere is formed by selecting a single point on the sphere and removing all points withing a certain distance, say $\delta$, from that point while keeping points with the exact distance $\delta$ to form a boundary.