Submersion (mathematics)

Submersion is a big word that mathematicians use to describe something very simple: like swimming underwater! Just like how your body is surrounded by water when you are swimming, mathematicians talk about something being surrounded by numbers or math things.

When we talk about submersions in math, we’re talking about a special kind of function that’s like a map. And just like how you follow a map to find where you’re going, we can use submersions to help us figure out things we need to know about numbers or shapes.

Here’s how it works. Imagine you have a big pile of shapes, and you want to know which shapes are the same. You could start by comparing each shape to the others one at a time, but that would take forever! Instead, you can use a submersion to find out which shapes are the same.

A submersion is a type of function that takes different shapes or numbers as input, and tells you which ones are the same. It’s like a rule that helps you sort out the shapes into groups. The special thing about a submersion is that it keeps the “structure” of the shapes the same. This means that if two shapes are “the same” according to the submersion, then they won’t just look similar - they’ll have the exact same shape or structure.

So, how can we use this idea to help us find shapes? Well, imagine you have a pile of shapes that all look like they have a “top” and a “bottom”. You could use a submersion to sort them out based on their tops and bottoms. The submersion would take the shape as input, and return the top and bottom, telling us if two shapes are the same based on their top and bottom.

Overall, submersion in math is a way to use functions to sort and organize things like shapes and numbers based on their structure. It’s a bit like swimming underwater, where you’re surrounded by water (or math things) and trying to find your way by following a map!