Well; when I looked at the key; I noticed it was a different key than the one I needed for that door.
That means I noticed it was different: that includes it's topology: it was a different key.
How did I "search" the patterns in the topology? Well; they key I needed was a flatter key, for starters.
Even if they were both locks for similar flat keys; they would have a different pattern of notches.
In a network of definitions: topolgy is identified by looking at the network from a range of perspectives. Presumably any group-perspective will throw up a complex topolgy that allows a detailed search for other ways of looking at the group.
This seems to be at the core of modern physics: finding every way of seeing a group perspective.
A quick probably messy maybe errors idea:
In quantum electrodynamics: they look with a group perspective (by using numbers and a definition by probability e.g. a dice is by definition 6 faces in 1 dice): so thay might look at a dice as 2 ways of seeing 6 faces to give a probability distribution which is still a definition.
They then interact the dice with something else (that is also defined in a similar way). They then work out all the ways the group-definitions can interact (some may cancel: as two possible strategies in a Chess game may have common moves and other moves that only can fit one of the strategies).
They get a final group view of how the groups can happen and still conserve their initial defining ingredients intact. They then call this final group view "the probability of an event happening" regarding a particular localised combined group view; but actually they just projected a definition of the event like a hologram, from their original definitions?
So topology was projected by interacting topologies and seeing what could survive.
A Chess player "searches the topology of game-strategy possibilities" by grouping; by linking desired move patterns into networks and considering ways of obtaining those networks; I guess.