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Topology

Topology

In the following, the terms of openness and closure of sets are reduced to absurdity. The Set Theory is presupposed.

Definition: 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 \(b\). In particular, an element \(x \in A \subseteq X\) is called neighbour of an element \(y \in A\), where \(x \ne y\) if we have for all \(z \in X\) and a mapping \(d: {X}^{2} \rightarrow \mathbb{R}_{\ge 0}\): (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 \(R, V \ne \emptyset = R \cap V\).

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 we apply \(\partial\) successively beyond that, we assume the argument to be without complement. The set \(A ° := A \setminus \partial A\) is called the interior of \(A\).

Definition: A set \(S \subseteq R \; (V)\) is said to be connected if we have 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 we have: 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.

Definition: Let \(\mathcal{P}(X) := \{A : A \subseteq X\}\) be the power set of the set \(X\). 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 \(B\).

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: Every \(U \subseteq R\) is called neighbourhood of \(x \in R\) if \(x \in U°\). A function between two topological spaces is said to be continuous if we have for every point: 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.

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. When \(a = 0\) and \(r = 1\), we obtain the unit ball with the special case of the unit disc \(\mathbb{D}\) for \(\mathbb{K} = \mathbb{C}\) and \(n = 1\).

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.

Definition: The function \(||\cdot||: \mathbb{V} \rightarrow {}^{(\omega)}\mathbb{R}_{\ge 0}\) where \(\mathbb{V}\) is a vector space over \({}^{(\omega)}\mathbb{K}\) is called a norm, if for all \(x, y \in \mathbb{V}\) and \(\lambda \in {}^{(\omega)}\mathbb{K}\), we have that: \(||x|| = 0 \Rightarrow x = 0\) (definiteness), \(||\lambda x|| = |\lambda| \; ||x||\) (homogeneity), and \(||x + y|| \le ||x|| + ||y||\) (triangle inequality). The dimension of \(\mathbb{V}\) is defined as the maximal number of linearly independent vectors, and is denoted by dim \(\mathbb{V}\). The norms \({||\cdot||}_{a}\) and \({||\cdot||}_{b}\) are said to be equivalent if there exist non-infinitesimal \(\sigma, \tau \in {}^{c}\mathbb{R}_{>0}\) such that, for all \(x \in \mathbb{V}\), it holds that:\[\sigma||x||{}_{b} \le ||x||{}_{a} \le \tau||x||{}_{b}.\]Theorem: Let \(N\) be the set of all norms in \(\mathbb{V}\). Every norm on \(V\) is equivalent if and only if \({||x||}_{a}/{||x||}_{b}\) is finite but not infinitesimal for all \({||\cdot||}_{a}, {||\cdot||}_{b} \in N\) and all \(x \in \mathbb{V}^{*}\).

Proof: We set \(\sigma := \text{min }\{{||x||}_{a}/{||x||}_{b}: x \in \mathbb{V}^{*}\}\) and \(\tau := \text{max }\{{||x||}_{a}/{||x||}_{b}: x \in \mathbb{V}^{*}\}.\square\)

Definition: The function \({\mu}_{h}: A \rightarrow \mathbb{R}_{\ge 0}\) where \(A \subseteq {}^{(\omega)}\mathbb{C}^{n}\) is an \(m\)-dimensional set with \(h \in \mathbb{R}_{>0}\) less than or equal the minimal distance of the points in \(A, m \in {}^{\omega}\mathbb{N}_{\le 2n}\), \({\mu}_{h}(A) := |A| {h}^{m}\) and \({\mu}_{h}(\emptyset) = |\emptyset| = 0\) is called the exact h-measure of \(A\) and \(A\) is said to be h-measurable. Let the exact standard measure be \({\mu}_{d0}\). If it is clear this is the standard measure, we may omit \(d0\).

Remark: This answers the measure problem positively: \({\mu}_{h}(A)\) is clearly additive and uniquely determined, i.e. if \(A\) is the union of pairwise disjoint \(h\)-homogeneous sets \({A}_{j}\) for \(j \in J \subseteq \mathbb{N}\), then\[{{\mu }_{h}}(A)=\sum\limits_{j \in J}{{{\mu }_{h}}\left( {{A}_{j}} \right)}.\]It is also strictly monotone, i.e. if \(h\)-homogeneous sets \({A}_{1}, {A}_{2} \subseteq {}^{(\omega)}\mathbb{K}^{n}\) satisfy \({A}_{1} \subset {A}_{2}\), then \({\mu}_{h}({A}_{1}) < {\mu}_{h}({A}_{2})\). If \(h\) is not equal on all considered sets \({A}_{j}\), we choose the minimum of all \(h\) and homogenise as described in Set Theory. This measure is more precise than other measures and is optimal, since its value is neither smaller nor greater than the distances of points parallel to the coordinate axes, as it simply considers the neighbourhood relations of points. Concepts such as \(\sigma\)-algebras and null sets are not required, since the only null set in this context is the empty set \(\emptyset\).

Examples: Consider the set \(A \subset [0, 1[\) of points, whose least significant bit is 1 (0) in their (conventionally) real binary representation. Then \({\mu}_{d0}(A) = \frac{1}{2}\). Real numbers represent a further refinement of the conventionally real numbers, by dividing the conventionally real intervals into (significantly) finer sub-intervals. Since A is an infinite (conventionally uncountable) union of individual points (without the neighbouring points of [0, 1[ in \(A\)) and these points are Lebesgue null sets, \(A\) is not Lebesgue measurable, however it is exactly measurable. Similarly, consider the subset \(Q\) of [0, 1[ \(\times\) [0, 1[ of all points with least significant bit 1 (0) in both coordinates. This set has exact measure \({\mu}_{d0}(Q) = \frac{1}{4}\).

Remark: A disk without its boundary conventionally represents an open set, because then each point of it has a conventional neighbourhood that lies completely in this set. Here, the idea is based on that, if the points are considered on a half-line, starting at the centre of the disk, it must always be considered a (real) neighbourhood for each point on this half-line towards the boundary.

In fact, however, "the end of the flagpole" must sometime be reached. So there must be a point in the interior of the disk, which has no conventional neighbourhood in this interior. Therefore, the term openness for sets is inept. If the unit disk is considered around the origin of ordinates, so the last point of the half-line \([0, 1[\), dually represented, is the point \(0.\overline{1}_{2}\) and the next point is the boundary point 1.

Between these two points lies no other point. The former has no neighbourhood that lies in the interior of the disk, though it is an interior point. For this reason, the disk without boundary is also closed, since the last points of the half-line form, beginning from the centre of the disk, just the closure as boundary. Since the neighbourhoods do not exist on their boundary, the term closure is meaningless for sets in Euclidean space.

Therefore it holds that at least in Euclidean space every open set is also closed, what reduces these terms to absurdity. This has not been further annoying, since infinitesimal quantities so far were not considered in a differentiated manner, thus in particular the numbers 1 and \(0.\overline{1}_{2}\) were equated. However, this is incorrect as is explained in Set Theory, since otherwise algebraicity (1) and transcendence \((0.\overline{1}_{2})\) are equated.

The absurd can also be illustrated by the fact that an infinite intersection of open sets such as all open concentric disks can form a closed set (the common centre of the disk). An infinite union of closed sets again can build an open set as an open disk does, as a union of all of its points as closed sets.

A 0-dimensional set (point) is therefore open, because every neighbourhood, also consists of one point. Hence, the empty set \(\emptyset\) is also closed, and as a consequence the whole Euclidean space is closed. Using spheres, that what has been said can easily be generalised to higher dimensions. The terms of the inner and outer point remain meaningful however, if any infinitesimal radiuses are permitted.

Since an absurd and meaningless special case also makes the general case here absurd or meaningless, one also for metric and topological spaces openness or closure of sets should not be considered, particularly since the definition of a conventional topological space appears oddly content-free and arbitrary, while the terms of interior and exterior point as well as boundary point are still useful and appropriate.

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 (imprecise) conventional mathematics.

© 2013-2019 by Boris Haase


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