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 \(N \subseteq {A}^{2}\) defines a NR in \(A \subseteq X\) for the underlying set \(X\).\(\triangle\)

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: 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: 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 \(r > \iota\) every real \(n\)-sphere, where \(n \ge 2\), is only connected and every real 1-sphere is not connected. It holds that \(r_1 = r_2 = 3.\)^{1}Gelbaum, Bernard R.; Olmsted, John M. H.: *Counterexamples in Analysis*; Republ., unabr., slightly corr.; 2003; Dover Publications; Mineola, New York., p. 160

Theorem: 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* \({}_{\tilde{\iota}}n\) if its elements consist of \(n\) maximal cubes of edge length \(\iota.\triangle\)

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^{2}see 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 topology^{3}cf. Kowalsky, Hans-Joachim: *Topologische Räume*; 1. Aufl.; 1961; Birkhäuser; Basel, p. 62 ff.. The situation is, however, different in partially imprecise conventional mathematics.

© 2013-2019 by Boris Haase

References

↑1 | Gelbaum, Bernard R.; Olmsted, John M. H.: Counterexamples in Analysis; Republ., unabr., slightly corr.; 2003; Dover Publications; Mineola, New York., p. 160 |
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↑2 | see Querenburg, Boto von: Mengentheoretische Topologie; 3., neu bearb. u. erw. Aufl.; 2001; Springer; Berlin, p. 83. |

↑3 | cf. Kowalsky, Hans-Joachim: Topologische Räume; 1. Aufl.; 1961; Birkhäuser; Basel, p. 62 ff. |