In the following section, Set Theory is presupposed.

Definition: A family of sets \(\mathbb{Y} \subseteq \mathcal{P}(X)\) is called topology on \(X \subseteq R\) if every intersection and union of sets of \(\mathbb{Y}\) belongs apart from \(\emptyset\) and \(X\) to \(\mathbb{Y}\). The pair \((X, \mathbb{Y})\) is called topological space. If \(\mathbb{Y} = \mathcal{P}(X)\), the topology is called discrete. A set \(B \subseteq \mathbb{Y}\) is called a base of \(\mathbb{Y}\) if every set of \(\mathbb{Y}\) can be written as union of any number of sets of \(B\). Every irreflexive relation \(N \subseteq {A}^{2}\) defines a neighbourhood relation in \(A \subseteq X\) for the underlying set \(X\). If \((a, b) \in N\), \(a\) is called neighbour of or neighbouring to \(b\).\(\triangle\)

Examples: The base for \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{A}_\mathbb{R}, \mathbb{A}_\mathbb{C}, \mathbb{R}\) and \(\mathbb{C}\) is precisely each related discrete topology.

Definition: Particularly, an element \(x \in A \subseteq X\) is called neighbour of an element \(y \in A\), where \(x \ne y\) if for all \(z \in X\) and a mapping \(d: {X}^{2} \rightarrow \mathbb{R}_{\ge 0}\) holds: (1) \(d(x, y) \le \text{max }\{\text{min }\{d(x, z), d(z, x)\}, \text{min }\{d(y, z), d(z, y)\}\}\) and (2) \(d(z, z) = 0\). Here \(d\) is called neighbourhood metric. Let \(P = R \cup V\) be the set of all points partitioned into actual points \(R\) and virtual points \(V\) for \(R, V \ne \emptyset = R \cap V\).\(\triangle\)

Definition: The set \(A’ := R \setminus A\), where \(A \subseteq R\), is called complement of \(A\) in \(R\). When \(R\) is clear from context, it can be omitted and \(A’\) can be called the exterior of \(A\). The set \(\partial V \; (\partial A)\) consists of all points of \(V \; (A)\) that have a neighbour in \(R \; (A’ \cup V)\), and is called the (inner) boundary of \(V \; (A)\). Here \(‘\) takes precedence over \(\partial\). When \(\partial\) is applied successively beyond that, the argument is assumed to be without complement. The set \(A ° := A \setminus \partial A\) is called the interior of \(A\).\(\triangle\)

Definition: A set \(S \subseteq R \; (V)\) is said to be connected if there is for every partition of \(S\) into \(Y \cup Z\) such that \(Y, Z \ne \emptyset = Y \cap Z\): \(\partial Y’ \cap \partial Z \ne \emptyset \ne \partial Z’ \cap \partial Y\). \(S \subseteq R\) is moreover said to be simply connected if holds: Both \(\partial Y’ \cap \partial Z \cup \partial Z’ \cap \partial Y\) for every partition into connected \(Y\) and \(Z\) and \(S’ \cup (\partial)V\) for \( S’\) as complement of \(S\) in \(R\) are connected for a connected (\(\partial)V\). Let \(P\) and \(R\) be simply connected. Every \(U \subseteq R\) is called neighbourhood of \(x \in R\) if \(x \in U°.\triangle\)

Definition: An \(h\)-homogeneous subset of \(R := \mathbb{R}^{m}\) for \(m \in \mathbb{N}^{*}\) is \(n\)-dimensional, where \(m \ge n \in \mathbb{N}^{*}\), if and only if it contains at least one \(n\)-cube with edge length \(h \in \mathbb{R}_{>0}\) and maximum \(n\). The definition for \(R := \mathbb{C}^{m}\) is analogous. Let be dim \({}^{(\omega)}\mathbb{C} = 2\). The set \({\mathbb{B}}_{r}(a) := \{z \in K := {}^{(\omega)}\mathbb{K}^{n} : ||z – a|| \le r\}\) for \(\mathbb{K} = \mathbb{R} \; (\mathbb{C})\) is called real (complex) (2)n-ball or briefly ball with radius \(r \in {}^{(\omega)}\mathbb{R}_{>0}\) around its centre \(a \in K\) and its boundary is called real (complex) (2)n-sphere \({\mathbb{S}}_{r}(a)\) or briefly sphere.\(\triangle\)

Examples: Every ball is simply connected and for \(r > d0\) every real \(n\)-sphere, where \(n \ge 2\), is only connected and every real 1-sphere is not connected. The hairy ball theorem does not hold:

Theorem: Strictly speaking, considering up to nine cells of a tic-tac-toe grid creates the singleton the only connected set with fixed point property.\(\square\)

Definition: When \(a = 0\) and \(r = 1\), the unit ball is obtained with the special case of the unit disc \(\mathbb{D}\) for \(\mathbb{K} = \mathbb{C}\) and \(n = 1\). A function between two topological spaces is said to be continuous if for every point that can be mapped holds: for every neighbourhood of the image of this point there is a neighbourhood of the point whose image lies completely in the neighbourhood of the image of this point. A set has dimension \({}_{\varsigma}n\) if its elements consist of \(n\) maximal cubes of edge length d0.\(\triangle\)

Theorem: For \(n \in \mathbb{N}_{\ge 2}\), no finite decomposition of an \(n\)-ball can be reassembled giving an \(n\)-cube, since finitely many convex boundaries cannot have the same order of concave counterparts.\(\square\)

Remark: For all contracting deformations, points are to be removed from the target set as the contraction specifies. Only then the (generalised) Poincaré conjecture holds.

Remark: The neighbouring boundary points of the conventional closed [0, 1] and the conventional open ]0, 1[ especially have not the Hausdorff property. So not every metric space can be a Hausdorff space or normal and (pre-) regular spaces are limited. The spaces \(\mathbb{C}^{n}\) and \(\mathbb{R}^{n}\) with \(n \in {}^{\omega }\mathbb{N}^{*}\) have therefore only the Fréchet topology. The situation is, however, different in partially imprecise conventional mathematics.

© 2013-2019 by Boris Haase