# Topology

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 $$R, V \ne \emptyset = R \cap V$$. 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 $$\{d(x, y) : x \in A°, y \in A^{\prime}\} = \hat{\nu}$$, 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 $$Y, Z \ne \emptyset = Y \cap Z$$: $$\partial Y’ \cap \partial Z \ne \emptyset \ne \partial Z’ \cap \partial Y$$. $$S \subseteq R$$ 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. It holds that $$r_1 = r_2 = 3$$1Gelbaum, Bernard R.; Olmsted, John M. H.: Counterexamples in Analysis; Republ., unabr., slightly corr.; 2003; Dover Publications; Mineola, New York., p. 160.

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 (invalid hairy ball theorem).$$\square$$

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.