# Metric Spaces

## Analogy

### Metric spaces are like intuitive maps.

When we give each other directions, we often speak in terms of the familiar. The maps we draw for each other are subjective. We omit and add information about how to get from one place to another based on where we are and who we are talking to.

Metric spaces are sets combined with a specific distance measure called a *metric*. For example, you could measure the distance from one coffee shop to another in terms of the distance a taxicab drives to get from one to the other.

In this case, the setwould include two points, the two coffee shops, and the distance measure used would be the taxicab distance.

## Drawing

### Metric spaces are visual by nature.

A metric space is a set combined with a distance measure.

TODO: interactive simulation of p-adic space

## Example

### The real line as a metric space

The real line is a metric space with the distance function $d(x,y)=|x-y|$

## Plain Description

### Metric spaces are maps that place similar items next to each other.

Metrics are distance measures that describe similarity. You can describe similarity using physical distance or any other distance measure that satisfies these qualities:

- never be negative
- positive distance between two distinct points
- distance between two distinct points doesn't change depending on direction of measurement
- satisfy the triangle inequality, that is: the distance from point A to point B is never larger than the distance from point A, to a point between them (like point C), to point B

## Technical

### Every distance measure satisfies four constraints.

$$d(x,y)\ge 0$$$$d(x,x)=0$$

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

$$d(a,b)\le d(a,c)+d(c,b)$$

the distance from point A to point B is never larger than the distance from point A, to a point between them (like point C), to point B