Completely metrizable space

Ok kiddo, do you remember what a metric is? It's like a rule that tells us how to measure distances between points. Well, a completely metrizable space is a special kind of space where we can measure distances in a very precise way, and we can do it using a certain kind of metric that has some extra properties.

Here's what it means: imagine you have a space, like a room, with a bunch of points in it. We want to be able to measure how far apart these points are from each other. To do that, we can use a metric – a formula that takes two points and gives us a number that represents the distance between them. For example, in a room, we might use the distance between two points as the length of the shortest path between them (like when we measure how far away something is with a ruler).

Now, in some spaces, there are different metrics we could use to measure distances. But in a completely metrizable space, we have a very special kind of metric that works for all the points in the space, and it has some extra properties that make everything work very nicely.

One of the most important properties of a completely metrizable space is that it's very "nice" geometrically – that is, it behaves just like we would expect a space to behave. For example, it's easy to imagine drawing straight lines or curves between points, and we can tell when two points are "close" or "far away" from each other. This makes it very useful for studying all kinds of things, like geometry, topology, and analysis.

So, to sum up: a completely metrizable space is a special kind of space where we can measure distances in a very precise and consistent way, using a special kind of metric that has some extra nice properties. This makes it a very useful tool for understanding all kinds of things about geometry and space, and it's also just really cool!