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The following section explains why the concepts of openness and closedness of sets should be rejected.

A disc without its boundary is traditionally considered to be an open set, since each of its points has a neighbourhood fully contained in the set. This is based on the idea that if we consider the radius from the centre of the disc passing through this point, each point in the radius has a neighbourhood of real points also contained in the radius.

But in truth the "end of the flagpole" must eventually be attained at some point. There must be some point in the disc without a neighbourhood in the interior of the disc. The concept of openness of sets is therefore flawed. If we consider the unit disc around the origin, the last point of the radius [0,1[, represented in binary form, is the point 0.1, and the next point is the boundary point 1.

There are no other points between these two points. The former does not have a neighbourhood in the interior of the disc, even though it is an inner point. Therefore, the disc without a boundary is also closed, since the last point of the radius from the centre of the disc is precisely given by its closure, the boundary. Working from the outside, it is easy to show that the concept of closedness is also meaningless for sets in Euclidean space, since their boundary points do not have neighbourhoods.

Analogously, it is easy to show that every open set of Euclidean space is simultaneously closed, which reduces these concepts to a contradiction. This has previously not posed an issue, since infinitesimal quantities have not been considered separately, and so in particular the numbers 0.1 have been viewed as equal. However, this approach is flawed, as explained in the chapter on Set Theory, as this amounts to equating algebraic (1) to transcendental (0.1).

The contradiction may also be shown by noting that an infinite intersection of open sets, such as the intersection of all concentric discs, each of which is open, can give a closed set (the centre common to all discs), and an infinite union of closed sets can create an open set, e.g. the open disc can be constructed as the union of all of its points, each of which represents a closed set on its own.

A set consisting of a single point is open because every neighbourhood must also consist of a single point, since the single-point set is 0-dimensional and intervals are one-dimensional. The empty set is therefore also closed, and consequently so is the entire Euclidean space. By arguing with balls, these statements may be easily generalised to higher dimensions. However, the concepts of inner and outer points still make sense when arbitrary infinitesimal radiuses are allowed.

Since the existence of an absurd or meaningless special case implies that the general case must also be absurd or meaningless, openness and closedness of sets in metric and topological spaces are not useful concepts. The definition of a topological space is oddly vacuous and arbitrary, whereas the concepts of inner and outer points and boundary points remain meaningful and useful.

The neighbouring boundary points of [0, 1] and ]0, 1[ (see Nonstandard Analysis) in particular do not have the Hausdorff property. Therefore, a metric space is not necessarily a Hausdorff space, and symmetric and (pre-)regular spaces are also restricted. ℂn and ℝn with n ∈ cℕ* are therefore only Kolmogorov spaces. Any stronger properties are only valid in (imprecise) conventional mathematics.

© 2013 by Boris Haase

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