## Topology

Outside of a dog, a book is a man’s best friend.  Inside of a dog, it’s very dark.

—Groucho Marx

Directions

Sometimes a precise equation gives you more information than you need, more than you asked for.  What if you asked me, “How do I get from Indianapolis to Chicago?”  I could answer by giving a precise series of (x,y) coordinates that one must traverse — like a table of all the (Latitude, Longitude) coordinates one passes through on the way.  But more likely you just want to know to take 465 until exit X, then 65 northwest until exit Y, and then 90 west until you reach the Chicago part of your directions.

Similarly if you wanted to get from Indianapolis to Guatemala City, I could just say:  take Megabus to Chicago, take the “El” train (blue line) to O’Hare, and fly TACA to La Aurora airport in Guate Guate La Capi.

$\large \dpi{200} \bg_white \rm{Indy} \overset{ \rm{Megabus}} \longrightarrow \rm{Chicago} \overset{\rm{El}} \longrightarrow \rm{O'Hare} \overset{\rm{TACA}} \longrightarrow \rm{La\ Aurora}$

Basically, you just want to know how to connect the parts of your route.  You want a topological kind of answer.

Topology Disregards Distance

Topological thinking dispenses with facts like “It’s 180 miles (4 hours) from Indy to Chicago” and just looks at the connections between things.  Am I inside or outside the house?  Is the storm going to evolve into a hurricane, or will it dissipate?  (Dynamical systems question.)  Is the economy on a path to sustainable growth, or mired in a self-destructive “spiral”?  (Again, dyn sys.)  How many ways are there to cross from the Minnesota Public Radio building to the parking lot?  (Minneapolis has enclosed skyways to keep you from freezing while you cross the street.)

(an 8_5 knot)

If you disregard distance and think that completely through, eventually you find yourself saying strange things like “A donut is topologically equivalent to a coffee cup” (they both have one hole).  Sometimes you want to know how far it is to Chicago, sometimes you want to know how many possible routes you can take to go there.  (If you counted all the county roads, switchbacks, and detours, that monstrous entanglement would have a LOT of holes.)

That’s when you use topology.

ANATOMICAL QUANDARY:  When you eat food, does it really go “inside” you?  After all, it hasn’t traversed a boundary—you have an open path straight through you from mouth to anus (it just temporarily closes the open loop with sphincters and stuff, from time to time).  To really be “inside” someone would be like somewhere between the lining of the stomach, and the skin.

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