Survey about Generalizing Distances

As known, in general topology the talking be about “nearness”. This is exactly needed to discuss subjects such convergence and continuity. The simple way to study about nearness is to correspond the set with a distance function to inform us how far apart two elements of are. The metric concept introduced by a French mathematician Maurice René Fréchet (1878 – 1973) in 1906 in his work on some points of the functional calculus. However, the name is due to a German mathematician Felix Hausdorff (1868 –1942) who is considered to be one of the founders of modern topology. In addition to these contribution, he contributed significantly to set theory, descriptive set theory, measure theory, and functional analysis.

Moreover, as the dimensionality increases of your data, the less useful Euclidean distance becomes. This has to do with the curse of dimensionality which relates to the notion that higher-dimensional space does not act as we would, intuitively, expect from 2-or 3-dimensional space. " [3] Manhattan distance is the sum of absolute differences between points across all the dimensions (̃,̃) = ∑ | − | =1 . "When your dataset has discrete and/or binary attributes, Manhattan seems to work quite well since it takes into account the paths that realistically could be taken within values of those attributes. Take Euclidean distance, for example, would create a straight line between two vectors when in reality this might not actually be possible." [4] Euclidean distance between ( 1 , 2 )and ( 1 , 2 ) Manhattan distance between ( 1 , 2 )and ( 1 , 2 ) Minkowski distance is the generalization of Euclidean and Manhattan distances

Pafnuty Lvovich Chebyshev Hermann Minkowski
"The upside to is the possibility to iterate over it and find the distance measure that works best for your use case. It allows you a huge amount of flexibility over your distance metric, which can be a huge benefit if you are closely familiar with and many distance measures" [5][6][7]. The limiting case → ±∞, yield the Chebyshev distance As mentioned before, Chebyshev distance can be used to extract the minimum number of moves needed by aking to go from one square to another. Moreover, it can be a useful measure in games that allow unrestricted 8-way movement", see [5][6][7] Minkowski's distance up on p Chebyshev's distance To imagine Minkowski's distance, follow drawing types of unit balls up on .

Unit ball up on
Finally, Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. As example, • 0000 and 1111 is 4. • 2173896 and 2233796 is 3.
"Typical use cases include error correction/detection when data is transmitted over computer networks. It can be used to determine the number of distorted bits in a binary word as a way to estimate error. Moreover, you can also use Hamming distance to measure the distance between categorical variables" [4].
As is known, the above distances have been developed to the state of the infinite dimension, where their effect in spaces like normed, Hilbert … etc. was clear. This great importance of metric distances encouraged researchers to find new generalizations to define the distance function itself some will be mentioned here. Firstly, (1) Every quasi-metric has a conjugate quasi-metric. The function ρ¯ defined by ρ¯(̃, ̃) = ρ(̃, ̃) for all ̃, ̃∈ Ω is also a quasi-metric on Ω and is called the conjugate quasi-metric of ρ. Also, the mapping ρ s (̃,̃) = max{ρ(̃, (2) In a quasi-metric space, the limit of a sequence is not necessarily uniqueness. [8

Definition (3): [9] A function
: where limiting point are defined in usual way. The space (Ω, ) by Frechet is called E-space and by Menger is called semi-metric.

Example (1) [10] Let
Various generalizations of standard distance have been studied. One of these distances is distance or ( −metric). It is the generalization of trigonometric inequalities which has been studied by Czerwik [11] Definition (4): [12]  Then (Ω, ) is a − metric space, but not a metric since the triangle inequality is not hold, i.e., On the other hand, we have the following example. " Example (3): [14] "Let (Ω, )be a metric space and (̃,̃) = ( (̃,̃)) where > 1. Then (Ω, )is a −metric with = 2 −1 . [15] In general, a −distance function is not continuous. It should be noted, −distance need not be jointly continuous in both variables. For this claim, the reader is referred to see the following example.
For many results and definitions Branciari's spaces, see [24,29,38,39,44,69]. Similar to the above generalizations Similar to the above generalizations, a generalization is given for the rectangular distance The following example shows that −rectangular distance not necessary rectangular.
And then the pair(Ω, )is called a 2 −metric space.
Remark (4): [38] "A 2 −metric is not a continuous function of its variables, every2 −metric is non-negative and it might assumed that every 2 −metric space contains at least three distinct points. " Remark (5): [39] Geometrically, Geometrically, (̃,̃,) is interpret as the area of the triangle spanned by ̃,̃ and . In Definition (1.1.17), if condition (iv) is canceled then is called a semi2−distance.
After that Dhage [40] gave another definition of distance together which is Definition (10): [40] "Let Ω be a nonempty set, and let denote the real numbers. Afunction : Ω 3 → satisfying the following axioms: "Unfortunately, most of the claims concerning the fundamental topological properties of −metric spaces are incorrect (see [41]). This claim provided inspiration for the formation of more general concept called a −distance by Mustafa and Sims [42]." then is called generalized distance on Ω and the pair (Ω, ) is called a − metric space . " Example (7): [43] "Consider Ω = Subsequently, many theoretical and applied studies appeared based on that distance. We will defer it to a later time, but you can see some of them as follows: [47][48][49][50][51][52].