/sci/ - Science & Math

>>11167285
>how is the mapping chosen/what equation/set of equations defines a map
It depends on how you parameterise (label) the points of your n-simplex. The standard definition (pic related) allows you to describe points by their n+1 coordinates. Then you can describe a mapping as a continuous function of those coordinates. I'm not going to actually write one out for you because it's pretty tedious (try it yourself)
In practice with singular homology there are uncountably many of these maps, so we need to use other techniques to actually compute homology groups, rather than directly talking about explicit maps.

>organizing bundled of elements of a field
Be careful what you mean here, a "field" has a precise mathematical meaning, which has nothing to do with topological spaces (as well as a different meaning in physics).

>no space at all required
Topos is literally the Greek word for space. Topology is an abstraction of the idea of space in the sense that it captures the notion of points being somehow related to each other. There's not really much point thinking about topological spaces until you've done basic analysis (metric spaces, etc). Before then the definition is incredibly opaque and doesn't seem motivated. Once you have done some analysis you see that often the basic properties of space which we actually rely upon are not distance or angles but open sets, and so it makes sense to forget about things like distance and angles and talk about spaces where you only have a notion of openness.

>Then it just shows up in various geometries depending how the field is graphically laid out.
???

>Also what are topological spaces with algebraic structure about
Think of for example the number line. The geometric properties of the number line behave well with addition/multiplication. Like if you have a number [math]x [/math] and add 1 to it, you get something close to what would happen if you picked a number close to [math]x [/math] and added 1 to that.