## 3 Materials and Methods
### Discrete Perfect Sets  
Suppose \(X\) is a finite set, and \(A,B,\dots\) and \(x,y,\dots\) are its subsets and points, respectively. **Definition 1**. : _Let us call a mapping of the product of \(2^{X}\times X\) into non-negative numbers \(\mathbb{R}^{+}\) in the set \(X\), increasing by the first argument and trivial-on-zero inputs as the density \(P\):_  
\[\begin{split} P(  
The \(A^{\infty}\) set demonstrates semi-invariance: its first density hull \((A^{\infty})^{1}\) does not fall beyond the \(A^{\infty}\) set.  
**Statement 2**.: \(A^{\infty}\) _contains its first \(\alpha\)-hull by the density:_  
\[(A^{\infty})^{1}\subseteq A^{\infty}\]  
Hence, it immediately follows that for a set \(A^{\infty}\) a series (2) of its \(\alpha\)-\(n\)-hulls is constant. **Consequence 1**. : \((A^{\infty})^{n}=(A^{\infty})^{1}\)__\(\forall n\geq 2\)__  
Let us designate \(\alpha\)-\(\infty\)-hull for \(A^{\infty}\) through \(A^{2\infty}\). Therefore:  
\[A^{2\infty}=(A^{\infty})^{\infty}=(A^{\infty})^{1}\subseteq A^{\infty}\]  
Sequentially plotting the \(\alpha\)-\(\infty\)-hulls based on the density \(P\), we obtain the following scheme:  
\[ \begin{array}{ccccccc}A&\rightarrow&A^{1}&\subseteq&\ldots&=&A^{\infty}\\ A^{\infty}&\supseteq&(A^{\infty})^{1}&=&\ldots&=&A^{2\infty}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ A^{m\infty}&\supseteq&(A^{m\infty})^{1}&=&\ldots&=&A^{(m+1)\infty}\\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\ \end{array} \]  
Due to the \(X\) finiteness in a non-increasing sequence  
\[A^{\infty}\supseteq\ldots\supseteq A^{m\infty}\supseteq\ldots\]  
starting with some number \(m^{*}\), stabilization occurs:  
\[A^{\infty}\supset\ldots\supset A^{m^{*}\infty}=A^{(m^{*}+1)\infty}=\ldots\]  
Let us designate the set \(A^{m^{*}\infty}\) by \(A(\alpha)\). The process of constructing \(A(\alpha)\) has a stage of increasing from \(A^{1}\) to \(A^{\infty}\) and a stage of decreasing from \(A^{\infty}\) to \(A(\alpha)\):  
\[A\to A^{1}\subset\ldots\subset A^{n^{*}}=A^{\infty}\supset\ldots \supset A^{m^{*}\infty}=A(\alpha) \tag{3}\]  
**Statement 3**. : \(A(\alpha)\) _matches its first \(\alpha\)-hull._  
**Remark 1**. : _Statement 3 means that the set \(A(\alpha)\) is comprised exactly of those points \(x\) of space \(X\) where its density is more than \(\alpha\) or equal to it:_  
\[A(\alpha)=\Big{\{}x\in X:P_{A(\alpha)}(x)\geq\alpha\Big{\}}.\]  
_Let us treat the density \(P_{A}(x)\) as a limiting measure of the point \(x\) for the set \(A\). The point \(x\) with sufficiently large density \(P_{A}(x)\geq\alpha\) is considered to be the limiting one for \(A\). Thus, the first hull of \(A^{1}\) represents the set of all limiting points for \(A\) in \(X\) in that sense. The points from the second hull \(A^{2}\) will, in general, be limiting points for \(A\) in \(X\) through \(A^{1}\)--that is, of the second order, etc. Statement 2 means that \(A^{\infty}\) is a closed set as it contains all its \(\alpha\)-limiting points in \(X\). Moving inside \(A^{\infty}\) leads to the already \(\alpha\)-perfect set \(A(\alpha)\), for restated Statement 3 means that \(A(\alpha)\) consists exactly of all finite points to it in \(X\), i.e., it is a perfect one._  
**Definition 3**. : _The set \(A\) consisting of exactly all \(\alpha\)-limiting with regard to this set points of the initial space \(X\) is called an \(\alpha\)-discrete perfect (simply perfect, DPS-) set in \(X\):_  
\[A\ -\text{DPS-set in }X\leftrightarrow A=\{x\in X:P_{A}(x)\geq\alpha\}\]Numerous studies and the examples below show that DPS-sets are condensations in \(X\) and are closely related to clustering therein. We have a way of generating them in \(X\), i.e., \(A\to A(\alpha)\) construction. It depends on four parameters: the initial space \(X\), set \(A\), density \(P\) and level \(\alpha\):  
\[A(\alpha)=A_{P}(\alpha|X).\]  
**Statement 4**. : _Dependencies for \(A\), \(P\) and \(X\) are increasing dependencies, and dependence for \(\alpha\) are decreasing dependence:_  
1. _If_ \(A\subseteq B\)_, then_ \(A(\alpha)\subseteq B(\alpha)\)_._
2.