homeomorphism

topology - noun
1 Mathematics the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.
• a family of open subsets of an abstract space such that the union and the intersection of any two of them are members of the family, and that includes the space itself and the empty set.
2 the way in which constituent parts are interrelated or arranged : the topology of a computer network.

So many parts of the October#110, 2004 interview of Pierre Huyghe provide fascinating insights into the complexity of his ‘dynamic’ works. Huyghe’s articulation against Baker’s questioning bring about the nature of certain aspects and reveal how he, Huyghe pushes his work through by way of an interest in mathematical systems - ‘topological systems’ –


PH: I am interested in an object that is in fact a dynamic chain that passes through
different formats. I am interested in a movement that goes through and
between some of the fields that you mentioned.

PH: […] I am more interested in what we can call topological systems.

PH: It is about how you use something. It refers to a process of translation.
However, when you translate something, you always lose something
that was in the original. In a topological situation, by contrast, you lose
nothing; it is a deformation of the same.

PH: It refers to an equivalence.

GB: […Philippe] Parreno writes:

Topology is concerned with the relative positions of figures, a question of
points, the set of which defines spaces. . . . A donut and a cup of coffee are
topologically equivalent because they describe the same space. An object
is a more or less complex situation which can be transformed into
another. By deforming it, by pushing it to its limits, we discover its affini-
ties with what exists outside of it.. . . To blow up an inner tube is to trans-
form it topologically.2

PH: It is the fold of a situation. It’s a way to translate an experience without
representing it. The experience will be equivalent and still it will be different.
(Baker/Huyghe, October#110, 2004. p.90-2)

Huyghe definitely has a knack for developing intricate systems or sets that utilize various ‘formats’ to tangle or maybe untangle inter-subjective relations(hips). It was easier for me to understand topological systems in a mathematical sense, as a family of open subsets than as an open/varied type of systematization explained by Huyghe within his art practice. One could place the open subsets as ‘formats’ or ‘fields’ they are singular yet become equivalent even when applied together or intersected they do not lose their characteristic but still retain this 'deformation' through possibly a form of isomorphic 'instance' - however that particular instance can 'open' out and this 'opening out' or 'outside itself' we could term it 'relation' becomes key.

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος (homoios) = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces — that is, they are the mappings which preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

Roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a doughnut are not. An often-repeated joke is that topologists can't tell the coffee cup from which they are drinking from the doughnut they are eating, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle.

Topology is the study of those properties of objects that do not change when homeomorphisms are applied. As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations (isomorphisms for instance) between them.
(http://en.wikipedia.org/wiki/Homeomorphism)

Maybe I'm wrong with my understanding of 'Topological Systems' but I think an attempt on my behalf to unravel/unpack its complex nature has strengthen my resolve to look at 'ideas' or art that certainly disrupts the norms whether they be of a 'relational' or 'situational' aesthetic is of little consequence. Its strength for me relies on its layers of complexity and 'opening up' of the work, those particular 'instances', in my humble opinion.