Franklin's Notes

Metric space

A metric space is a set $M$ equipped with a distance function $d:M\times M\to\mathbb R^{\geq 0}$ satisfying the following three axioms:

1. Positive definiteness: $d(x,y)\geq 0$, and $d(x,y)=0$ iff $x=y$.
2. Symmetricity: $d(x,y)=d(y,x)$.
3. Triangle inequality: $d(x,z)\leq d(x,y)+d(y,z)$.

These are meant to formalize our intuition of what "distance" means. Positive definiteness states that distance is always a nonnegative number and that a distance of zero indicates the "same location". Symmetricity indicates that the distance between two locations does not depend on the "direction of travel". And finally, if distance is taken to mean the length of the "shortest path" between two locations, the triangle inequality can be thought of a stating that visiting some intermediate location along the way cannot make your path any shorter, but it may make it longer.

If $d:M\times M$ satisfies only the axioms (1) and (2), then it comprises as semimetric space.

Given a sequence of points $(p_n)\subset M$, we say that it converges to some point $p\in M$ iff in which case we also write, as a notational convention,

Proposition 1. In a metric space, no sequence can have two distinct limits.

Proof. Suppose that $p,p'$ are both limits of the sequence $(p_n)\subset M$, so that $d(p,p_n)\to 0$ and $d(p',p_n)\to 0$ as $n\to\infty$. Then we have that for all $n\in\mathbb N$ by the triangle inequality, but since $d(p,p_n)+d(p',p_n)\to 0$ as $n\to\infty$, we have that $d(p,p')=0$ by the Squeeze Theorem. But by positive definiteness, this means that $p=p'$. $\blacksquare$




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