Pseudometric space

Imagine you have a big room where you can play with your toys. Now imagine that you have a special toy (let's call it a "ball") that you can use to measure the distance between any two other toys in the room.

However, this "ball" is not a regular ball. It behaves a little differently than what you might expect. For example, if you throw it really hard, it might come back to you instead of bouncing off a wall or object.

Now, let's say you want to use this "ball" to measure the distance between two of your toys. You would throw the "ball" from one toy towards the other toy and see where it ends up. Depending on how far away the other toy is, the "ball" might come back to you, or it might stop near that toy, or it might even go past that toy and keep going until it hits another object.

After you've done this a few times, you can start to get a sense of how far apart the toys are from each other, based on how the "ball" behaves when you throw it between them.

This is kind of like what a pseudometric space is. Instead of toys and a special "ball", we have a set of objects (which can be anything, like points in a graph or functions) and a way of measuring the distance between them, which is not necessarily the same as what you might expect from a regular "metric" (like the Euclidean distance between two points).

In a pseudometric space, the "distance" between two objects might not behave exactly like you think it should. For example, it might not satisfy the "triangle inequality", which says that if you want to go from object A to object C, passing through object B, then the distance from A to C should be no longer than the sum of the distances from A to B and from B to C.

But just like with your toys and your special "ball", we can still use these "distances" to understand how objects in the pseudometric space are related to each other, even if they don't behave exactly like regular distances would.