Originally Posted by

**ThePerfectHacker** Given a non-empty set $\displaystyle X$.

Such as function such as,

$\displaystyle d:X\times X\to \mathbb{R}^+$

which satisfies.

$\displaystyle d(x,y)\doublearrow x=y$

$\displaystyle d(x,y)=d(y,x)$

$\displaystyle d(x,y)\leq d(x,z)+d(z,y)$

Underthese condition we have a metric space $\displaystyle d$ on set $\displaystyle X$ denoted by $\displaystyle (X,d)$.

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The intuitive, concept is to capture the idea of distance for any sets. Cuz,

1)It is measured in real numbers (positive) like any distance.

2)Makes no difference which point is first and second.

3)Triangular inequality for distance.

I do not see any more basic way to explain it rather than saying its is distance defined for any sets.

(I hate these type of people that argue with known concepts that they are incorrect and yet themselves never fully studied them!)