Real projective space
Okay kiddo, imagine you're playing with a ball in a big playground. You can throw the ball in any direction, but when it hits the fence, it bounces back in the opposite direction. Now imagine that the playground is actually a big space that goes on forever in all directions, with no fences. If you throw the ball really far, it will just keep going and going without ever hitting anything.
But what if we add a special rule? Instead of bouncing back when it hits the edge of the space, the ball goes through the edge and comes out on the other side. So if you throw the ball far enough in one direction, it will eventually come back from the opposite direction, like if you could walk around the whole world and end up where you started.
Now let's apply this idea to something called real projective space. This is a special kind of space where points are identified with each other in pairs. That means if you have two different points, they might look different, but in real projective space, they're actually the same.
To help you understand this, imagine you're drawing pictures of animals. If you draw a picture of a cat, we can call that point "cat". But if you draw a picture of a dog, we can call that point "dog". In real projective space, we identify certain pairs of points as being the same. So if you draw a picture of a cat and a picture of a dog in the same place, we call that point "cat-dog".
Now comes the tricky part. Just like the ball bouncing through the edge of the playground, we have a special rule for how points in real projective space interact with each other. If two points are identified as the same, that means we can "glue" them together into one point. We call this process "projectivizing".
To help you picture this, imagine you have a picture of a cat and a picture of a dog glued together. If you projectivize those two points, you end up with a point that represents both a cat and a dog at the same time! It's like a hybrid animal.
Real projective space is a space where we identify certain pairs of points as being the same, and we have a special rule for how points interact with each other. We can projectivize pairs of points to create new, hybrid points. It's a strange and fascinating kind of space, but don't worry if it doesn't make complete sense just yet - even grown-ups need time to wrap their heads around these concepts!
But what if we add a special rule? Instead of bouncing back when it hits the edge of the space, the ball goes through the edge and comes out on the other side. So if you throw the ball far enough in one direction, it will eventually come back from the opposite direction, like if you could walk around the whole world and end up where you started.
Now let's apply this idea to something called real projective space. This is a special kind of space where points are identified with each other in pairs. That means if you have two different points, they might look different, but in real projective space, they're actually the same.
To help you understand this, imagine you're drawing pictures of animals. If you draw a picture of a cat, we can call that point "cat". But if you draw a picture of a dog, we can call that point "dog". In real projective space, we identify certain pairs of points as being the same. So if you draw a picture of a cat and a picture of a dog in the same place, we call that point "cat-dog".
Now comes the tricky part. Just like the ball bouncing through the edge of the playground, we have a special rule for how points in real projective space interact with each other. If two points are identified as the same, that means we can "glue" them together into one point. We call this process "projectivizing".
To help you picture this, imagine you have a picture of a cat and a picture of a dog glued together. If you projectivize those two points, you end up with a point that represents both a cat and a dog at the same time! It's like a hybrid animal.
Real projective space is a space where we identify certain pairs of points as being the same, and we have a special rule for how points interact with each other. We can projectivize pairs of points to create new, hybrid points. It's a strange and fascinating kind of space, but don't worry if it doesn't make complete sense just yet - even grown-ups need time to wrap their heads around these concepts!
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