Complete metric

Okay, so you know how we use rulers to measure how long something is? Well, imagine that we want to measure how far apart two places are. A complete metric is like a special ruler we use to measure this distance.

The complete metric is made up of something called a space, which is made up of points that we can measure the distance between. It's kind of like a big imaginary playground where each point is like a different toy you can play with.

Now, to measure the distance between two points, we use something called a distance function. This is basically a way to tell how far apart two points are. It's like when we say "my house is 3 miles away from the store" – the distance function tells us how many miles or how much space we need to go from one point to another.

But what makes a complete metric special is that it tells us exactly how far apart all the points are. It's like having a really accurate ruler for measuring distances. And it's called "complete" because it includes all the points in the space – there are no missing spaces or gaps where we can't measure the distance.

So when we say a space has a complete metric, we mean that we can measure the distance between any two points in that space, and we don't miss any points or spaces when we do so. That's important because it helps us understand how things are connected and how far apart they are from each other.