Monotonically normal space
Imagine you're playing with blocks and you stack them on top of each other. You start with a big block on the bottom and then you stack smaller blocks on top of it. The final tower of blocks starts big and gets smaller as it goes up.
In a monotonically normal space, you can think of the blocks like open sets (areas that are not occupied by anything). Each set contains the set below it, like how the smaller blocks fit inside the bigger blocks. The interesting thing about a monotonically normal space is that if you have two sets, one containing the other, and you want to find a continuous function that maps the bigger set to a smaller set, then you can do it in such a way that the pre-image (the part of the bigger set that maps to the smaller set) is contained entirely in the smaller set.
This might sound complicated, but it basically means that if you have a bigger open set, and you want to shrink it down to a smaller open set, you can do it in a way that doesn't overlap with any other open sets in the space. It's like taking the smaller block out of the tower without touching any of the other blocks.
So, a monotonically normal space is a special kind of space where you can shrink open sets in a very precise way, without messing up other parts of the space. It's a bit like playing a Jenga game, but instead of blocks, you're dealing with open sets, and instead of pulling them out randomly, you can only take out the parts that don't touch other open sets.
In a monotonically normal space, you can think of the blocks like open sets (areas that are not occupied by anything). Each set contains the set below it, like how the smaller blocks fit inside the bigger blocks. The interesting thing about a monotonically normal space is that if you have two sets, one containing the other, and you want to find a continuous function that maps the bigger set to a smaller set, then you can do it in such a way that the pre-image (the part of the bigger set that maps to the smaller set) is contained entirely in the smaller set.
This might sound complicated, but it basically means that if you have a bigger open set, and you want to shrink it down to a smaller open set, you can do it in a way that doesn't overlap with any other open sets in the space. It's like taking the smaller block out of the tower without touching any of the other blocks.
So, a monotonically normal space is a special kind of space where you can shrink open sets in a very precise way, without messing up other parts of the space. It's a bit like playing a Jenga game, but instead of blocks, you're dealing with open sets, and instead of pulling them out randomly, you can only take out the parts that don't touch other open sets.