Hole

Let D be a compact subset of the complex plane, and z₀ be a point not contained in D. The punctured compact set, denoted by D{z₀}, is defined as the set obtained by removing the point z₀ from D. Mathematically, we can express it as:

$$D{z₀} = {z \in \mathbb{C} : z \in D, z \neq z₀}$$

This notation indicates that the set D{z₀} consists of all points z in the complex plane such that z is in D and z is not equal to z₀.

In terms of formulas, if D is given by its boundary curve Γ, parametrized as Γ(t) = x(t) + iy(t), where t lies in some interval [a, b], the punctured compact set D{z₀} can be described using set-builder notation as:

$$D{z₀} = {z = x + iy \in \mathbb{C} : (x, y) = (x(t), y(t)), t \in [a, b], z \neq z₀}$$

Here, (x, y) represents the Cartesian coordinates of the complex number z, and (x(t), y(t)) denotes the parametric representation of points on the boundary curve Γ.

By defining a hole in this manner, we create a compact set with a missing point, which can have various implications and applications in mathematical analysis and the theory of Fredholm operators.