Day 13 - Up Like a Lead Rock

Phew, that was close. I just stumbled in from baby sitting next door, my eyes already tired. "I must sleep now," I thought, as I shambled upstairs, "I have to wake up at 5:30am again." Fortunately, my laptop screen was on, and the harsh glare stared at me accusingly.
(Can a glare stare? It can now.)

So, here I am. What did I do today? Drove Julia to work, slept, watched Zim, played Crossbow Training, watched Speed ("Spid"), read Frege, babysat. Why would you want to here about that though? Heck, I've only got seven minutes before I go to bed, and I'm not going to waste that precious time telling you all the little details.

Six minutes. A stitch in time saves nine. Oh darn. So what should I write about today?

Topology. Five minutes.

Topology deals with properties that hold of a shape (surface, space, object, etc) no matter how you stretch or squeeze it. Topologically speaking, a coffee cup is the same as a doughnut (the kind with a hole in the middle, that is). Mathematically speaking, this is done by giving a certain set (i.e. the space, all the points on the coffee cup, or doughnut, or sphere) and a certain structure on them.
This is how anything is given, really. A mug, regarded 'normally', is just a set of points with the structure "this bit goes there, this bit goes underneath it, ..." and so on.
Viewing it topologically, we're not concerned with all this elaborate structure; only the sort of structure that doesn't change when we squeeze, stretch, inflate, and so on.
This is done via 'open sets'. That is, a notion of 'closeness' that is preserved under stretching. When the surface is stretched, so are the open sets, so this closeness is preserved. Of course, poking a hole in the surface is not allowed, since this might change some open sets.

So, a topology on a set is a collection of sets, which we call the open sets, obeying certain axioms to make sure that we can't just have any old sets as the open sets. We can still have a lot though; especially remember that the same set can have many, infinitely many different topologies.

A topological space is a set with a topology, i.e. a collection of open sets. Topology deals with topological spaces.

0 minutes. Good night.