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# Topology

In the following section, Set Theory is presupposed with the Euclidean norm $$||\cdot||$$.

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 $$N \subseteq {A}^{2}$$ founds a NR in $$A \subseteq X$$ for the underlying set $$X$$.$$\triangle$$

Definition: If $$(a, b) \in N$$, $$a$$ is called neighbour of or neighbouring to $$b$$. 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 $$R, V \ne \emptyset = R \cap V$$. When $$R$$ or $$V$$ is clear from context, it can be omitted.$$\triangle$$

Definition: The set $$A^{\prime} := R \setminus A$$, where $$A \subseteq R$$, is called complement of $$A$$ in $$R$$. $$A^{\prime}$$ can be called the exterior of $$A$$. All points of $$V \; (A)$$ that have a neighbour in $$R \; (A^{\prime} \cup V)$$ form the (inner) boundary $$\partial V \; (\partial A)$$ of $$V \; (A)$$. Here $${}^{\prime}$$ 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 $$A$$. If $$\partial A \subseteq A$$ is increased by the condition min $$\{d(x, y) : x \in A°, y \in A^{\prime}\} = \tilde{\nu}$$, let $$A^{\ll} := A \setminus \partial A.\triangle$$

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 $$Y, Z \ne \emptyset = Y \cap Z$$: $$\partial Y^{\prime} \cap \partial Z \ne \emptyset \ne \partial Z^{\prime} \cap \partial Y$$. $$S \subseteq R$$ is moreover said to be simply connected if holds: Both $$\partial Y^{\prime} \cap \partial Z \cup \partial Z^{\prime} \cap \partial Y$$ for every partition into connected $$Y$$ and $$Z$$ and $$S^{\prime} \cup (\partial)V$$ for $$S^{\prime}$$ 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: If $$\emptyset \ne \mathbb{D} \subseteq (X, \mathbb{Y})$$ holds, a connected $$\mathbb{D}$$ is called domain. The set of $$h$$-$$S \subseteq \mathbb{R}^{m}$$ 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 $$n$$. Let $${}^1\dot{\mathbb{R}}^n$$ the unit ball with the special case unit disk $${}^1\dot{\mathbb{R}}^2$$. Midpoints $$a$$ of $$n$$-balls and $$n$$-cubes may be denoted in brackets as ($$a$$) after those. A set has dimension $${}_{\tilde{\iota}}n$$ if its elements consist of $$n$$ maximal cubes of edge length $$\iota.\triangle$$

Examples: The base for $$\mathbb{N}, \mathbb{Z}, \mathbb{A}_\mathbb{R}, \mathbb{A}_\mathbb{C}, \mathbb{R}$$ and $$\mathbb{C}$$ is precisely each related discrete topology. The boundary of every $$n$$-ball with $$n \ge 2$$ is only connected, itself it is simply connected. For $$n = 1$$ both are not connected. For $$n \ge 2$$ and $$r \in \mathbb{R}_{>0}$$, the $$n$$-Torus $${}^r\mathbb{T}^n := \left(\partial{}^r\dot{\mathbb{R}}\right)^n$$ is only connected.

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.$$\square$$

Theorem: Traversing up to nine cells of a tic-tac-toe grid renders the singleton the only connected set with fixed point property (invalid hairy ball theorem).$$\square$$

Definition: 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.$$\triangle$$

Remark: The suggestive terms compactness and countability (possibly misleading) are not used since they are not given for infinite sets like $$\mathbb{R}$$ and $$\mathbb{N}$$ resp. 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 property1see Querenburg, Boto von: Mengentheoretische Topologie; 3., neu bearb. u. erw. Aufl.; 2001; Springer; Berlin, p. 83. 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 topology2cf. Kowalsky, Hans-Joachim: Topologische Räume; 1. Aufl.; 1961; Birkhäuser; Basel, p. 62 ff.. The situation is, however, different in partially imprecise conventional mathematics.