Problem Analysis: Path Connected

May 1, 2017

Last time we talked about blobs that have holes in them.  This time lets talk about how we might deal with a blob that is hard to deal with because the valid points in it are hard to conceptualize (either because the blob has a lot of holes or for some other reason).

In topology (and by extension in metric spaces) there’s a concept called path connected.  The approximate idea is that any two points in the space will have a path in between them.  There’s a lot of implications here, but the one we care about is that now instead of needing a complex intuition about the blob we are worrying about (filled with complex holes and precarious boundary conditions) all we have to worry about is an equation that indexes all valid points in the blob.

What’s going on here is that we’re finding a safer way to conceptualize the blob that will make it easier to work with because there are less details that you have to worry about.  You might still lose out on a lot of intuition involving distance that you would normally have with a metric space with few holes, but there is still a lot of natural intuition that you can leverage from building a path between all of the points in your blob.

Problems with path connected blobs are easier to work with than problems with blobs that are not path connected.

Next time we’re going to move onto aspects of arrows.