About

Me

I’m zrkrlc (pronounced ˈzɜːkəlɒk (in British IPA) or ‘zer-ker-lock’ in approximate English). I don’t do stuff for a living yet (fortunately) so I write about whatever tickles my pickle. And yes, that sounds as wrong as it reads so don’t say it out loud.

My life’s accomplishments include:

  • 12 000+ hours of Internet use since 2007
  • turned eight without burning the house down
  • learned how to wiggle both my eyebrows
  • bought 36 jars of made-in-the-mountains strawberry jam on a single trip
  • got lost in a citadel for over two hours when I was in second grade

I also write code in Haskell, Python, and C# to varying degrees of proficiency.

This Website

I laid some prohibitions for myself in my first post, Things I will try NOT to do.

I will try to write as much as my time would allow. Being unemployed is a difficult commitment to keep so I try to let off steam here as much as I have to. Oh, and don’t worry: I don’t plan on inundating you with funsies all the time. Sometimes things have to get licking-your-face serious and I hope that doesn’t turn you off.

The Header Image

Pictured at the beginning of every page on this site is the topologist’s sine curve, so named because it’s a textbook example of why pedestrian’s don’t bother with such abstract nonsense. It’s the graph of the function f:(0, 1] \rightarrow \mathbb{R} plus the origin (0, 0) defined by f(x) = sin(1/x). If we define the following notions:

Fig. 1: Examples and non-examples of topological spaces. Note that these are sets. Illustration taken from Wikipedia1..

DEF: (topological space)
a set X of points x together with a function \mathbf{N}: X \rightarrow 2^{X} \setminus \emptyset whose elements we shall call the neighborhoods of x, such that:
  • x \in N whenever N \in \mathbf{N}(x)
  • If N_1 and N_2 are \in \mathbf{N}(x), then N_1 \cap N_2 is also \in \mathbf{N}(x).
  • If N \subseteq X and \exists S \in \mathbf{N}(x) such that S \subseteq N, then N \in \mathbf{N}(x).
  • (\forall N \in \mathbf{N}(x))(\exists M \subseteq N)(\forall a \in N)(M \in \mathbf{N}(a))
DEF: (connectedness)
a topological space is connected if it is inexpressible as a union of disjoint nonempty open sets
DEF: (neighborhood system)
the set of all neighborhoods of a point
DEF: (local connectedness)
a topological space is locally connected if there exists a neighborhood system for all points which are composed entirely of open, connected sets
DEF: (path)
a path f: [0, 1] \rightarrow X from a point x to a point y in a topological space X is a continuous function such that f(0) = x and f(1) = y
DEF: (pathwise connectedness)
a topological space X is pathwise connected if there exists a path between any two points in X

 

Then it can be shown that f is connected but neither locally connected nor pathwise connected. I like maths.

This also brings us to the title.

The Title

Fig. 2: A homotopy. GIF taken from Jim.belk2.

DEF: (homotopy)
Let X and Y be topological spaces and f and g mappings between them. A function H: X \times [0, 1] \rightarrow Y is a homotopy if (x \in X) \implies (H(x, 0) = f(x) \land H(x, 1) = g(x)).

 

Which basically means, if you pick up a path, beat it with a sledgehammer (without breaking it) and put it down again, then you just did a homotopy. Did I tell you I like maths?


1. From Wikipedia.

2. Also from Wikipedia.

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