Fig. 3

A line, square or cube all exist in Euclidean space with a certain number of dimensions described classically by D _E  = 0 for a single point, 1 for a line (a), 2 for a plane (b) and 3 for a 3D object (d) [38]. The concept of topology is rooted in the idea of connectedness among points in a set. The null (empty) set in topology (∅) has no points and its D _T is by definition ‘-1’. A single point or a number of points makes up a ‘countable set’. In topology, a set’s D _T is always 1 integer value greater than the particular D _T of the simplest form that can be used to ‘cut’ the set into two parts [42]. A single point or a few points (provided they are not connected) are already separated, so it takes ‘nothing’ (∅) to separate them. Thus the D _T of a point is 0 (−1 + 1 = 0). A line (a) or an open curve can be severed by the removal of a point so it has D _T  = 1. A topological subset such as b can have an interior, boundary and exterior. b has a closed boundary of points (like y). When its interior is empty, b is referred to as a boundary set. Its interior may instead be full of points (like x) that are not boundary points because separating them from the exterior is a neighbourhood of other points also contained in b. All points of the subset that are neither interior nor boundary will form the exterior of b. A line of D _T  = 1 is required to split this topological set into 2 parts, therefore the D _T of b = 2. Flat disks (c) have D _T  = 2 because they can be cut by a line with a D _T  = 1. A warped surface can be cut by a curved open line (of D _T  = 1) so its D _T  = 2 although its D _E  = 3. Therefore, while lines and disks have D _T  = D _E , warped surfaces have D _T one less than D _E . D _E = Euclidean dimension; D _T = topological dimension