Metric spaces: An abstract study of distance of elements in a set.



The word metric literally means distance, and metric spaces is the study of distance or a study of how close elements are to each other.
The study of metric spaces is attributed to the French mathematician Maurice Fréchet who in 1906 developed "abstract analysis" in his thesis where he first used abstract terminologies such as metric space, completeness, compactness etc.
A metric space is not just a set, in which the elements have no relation to each
other, but a set $X$ equipped with a particular structure, its distance function $d$. One can emphasize this by denoting the metric space by the pair $(X,d)$, although it is more convenient to denote different metric spaces by different symbols such as $X,Y,Z$.
Therefore, we define a metric space $X$ as a function $d:X\times{X}\rightarrow\mathbb{R}^+$ such that the following properties hold for all $x,y,z\in{X}$:
  1.  Triangular inequality: $d(x,y)\leq{d(x,z)+d(z,y)}$
  2.  symmetry: $d(y,x)=d(x,y)$
  3.  Identity: $d(x,y)=0\Leftrightarrow{x=y}$.          
A literal translation of the above axioms or properties can be given as:
  1.  the distance from point $x$ to point $y$ is less than or equal to the distance from point $x$ to $z$ via any third point $z$ to point $y$. 
  2. -the distance from point $y$ to point $x$ is the same as the distance from point $x$ to point $y$. 
  3.  If the distance from the points $x$ to $y$ is zero then it implies that $x$ and $y$ are the same points.

It is important to note that, $X$ denotes an abstract set with a distance, not necessarily $\mathbb{R}$ or $\mathbb{R}^N$. Elements contained in $X$ are referred to as points which may be geometric points, sequences or functions. Metric spaces unlike their vector space sister does not have the operations of "addition" and "scaler multiplication" while on the other hand vector spaces have a vector called the "zero vector"; in the special case of the vector space $k^n$ (where $k$ is a field), this vector is often called "the origin", since $k^n$ also can be seen as a geometric object (the $n$-dimensional affine space). But vector spaces don't necessarily have something we call "the origin": the collection of all polynomials with real coefficients is a real vector space, but we don't normally refer to the zero polynomial as "the origin", even though it is the zero vector of this vector space. So when we are trying to see the differences between a vector and metric we must also note that metric space is a set with a notion of distance defined between points of that set. This notion of distance is a function known as the metric (which must satisfy a set of axioms pertaining to distance). This metric takes in any two points and maps them onto a real number which characterises the distance between those two points.
A vector space is a set containing objects called vectors which interact in some pre-defined way determined by the axioms. Vectors are measured relative to some reference frame and thus have a notion of magnitude and direction from some origin. There are two additional properties satisfied by some metric spaces that merit particular attention:
  1.  The complete metrics, which guarantees Convergence of cauchy sequence. 
  2. Separable metric spaces, where elements in the metrics are handled by approximation.
The compact metrics spaces possess this two additional properties. 

Some examples of metric spaces 
  1.  Any subset of a metric space is itself a metric space as far as the axioms of metric space is valid for the points in the subset. 
  2.  The spaces $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$ and $\mathbb{C}$ have the standard distance $d(a, b) :=|a − b|$.
  3.  The vector spaces $\mathbb{R}^N$ and $\mathbb{C}^N$ have the standard Euclidean distance defined by $d(x,y):=\sqrt{\sum^{N}_{i=1}|a_i-b_i|^2}$ for $x=(a_1,..., a_N),y=(b_1,..., b_N)$.
  4.  The positive real numbers with distance function $d(x,y)=|\log(y/x)|$ is a complete metric space.
  5.  The hyperbolic plane is a metric space. 
  6.  An injective function $f$ from any set $A$ to a metric space $(X,d),d(f(x),f(y))$ defines a metric on $A$.

There are pages upon pages of examples on metric spaces but this the few we can mention for now.