Very Strange Characterisation of Topology Definition of Topology
I don't understand this characterisation of a topology, I don't even comprehend what $g\colon\{ \{\{\}\},\{\{\},\{\{\}\}\}\} \rightarrow T$ should mean? Can somebody explain it to me?
 A: This is a bit hard to read when formatted as a comment, but here is the "motivation" in the words of the Wikipedia user who made that particular edit in October 2012.
Originally the definition was short and plain.  Perhaps a bit quaint in the use of postfix, and subpar grammar and typography, but it isn't hard to piece together.  The idea is to define a topology in terms of $T$ first rather than as a collection of subsets of $X$, which is not an unreasonable idea.

A topology is a set $T$ such that
  
  
*
  
*For all set $I$.For all function $f:I\to T$ $\cup(i)f \in T$.
  
*For all function $f:\{1,2\} \to T$ $(1)f\cap(2)f \in T$.

This was subsequently deleted by another user who described it as "an attempt to define $T$ as set of open sets in a most unreadable manner."  The previous user then restored the content, adding the following message:

Hello,
First of all I want to say that the definition of a "topology" rather
  than a Topological space is standard (ie Kelley General Topology) The
  definition is correct and you can easily prove it The notation (a)f
  instead of f(a) is standard too and, imo, it emphasises the formal
  idea of a function, so to say (a)f is to say a in pi_1(f) where
  f:=(A,f,B) so, for example, b=(a)f is to say (a,b) in f. i corrected
  and supress the use of the non-necessary symbols 1,2

The current revision would appear to be merely a sarcastic response (temper tantrum?) to the criticism.  The part that convinces me is how thoroughly he/she eliminated the use of "non-necessary" symbols 1,2, even from the enumerate list!
A: It is just an attempt to express the usual axioms of topology in a more "formal" manner. Like Lord_Farin wrote, we can write the sets $\mathbf{0} = \emptyset = \{\}$, $\mathbf{1} = \{\mathbf{0}\} = \{\{\}\}$, and $\mathbf{2} = \{\mathbf{0}, \mathbf{1}\} = \{\{\}, \{\{\}\}\}$ using von Neumann's notation for integers. Then the domain of $g$ is just the two-point set $\{ \mathbf{1}, \mathbf{2}\}$. 

Now, $f:I \to T$ means that for each $i\in I$ we have $(i)f \in T$ is an element of the topology (here we follow the Wikipedia article to act functions on the right instead of on the left as usual). So $\cup_{I}(i)f \in T$ for any set $I$ and any function $f:I \to T$ is just a way to say, in notations, that "$T$ is closed under arbitrary union." 
On the other hand interpreting $g: \{\mathbf{1},\mathbf{2}\} \to T$ and $(\mathbf{1})g \cap (\mathbf{2})g \in T$ just say, in notations, that "$T$ is closed under pairwise intersections". 
A: The set $\{0,1\}$ with the Sierpiński topology, which I will call $\rm S$ has the following property : there is a bijection between the topology $\mathcal T$ of any topological space $\rm X$ and the continuous functions $\rm X \to \rm S$.
