The opening chapters of the book on algebraic topology I’m reading are concerned with setting up the basic notions of topological spaces, neighbourhoods, open sets and things like that. The key point is to move from a notion of space defined in terms of distance and proximity, a “metric space” where it is possible to measure the distance between one point and another, to a notion of space defined wholly in terms of the set-theoretic relations of belonging and inclusion. A neighbourhood of a point is no longer just a set of “nearby” points: it’s a set of points having a particular structure of inclusiveness (and includedness) in relation to other sets of points in the topological space.

In particular, this gives rise to a concept of space abstracted from our bodily experience of being orientated in a three-dimensional field and dealing with objects that are near to and far from ourselves and each other. It may be that the concept of topological space stands as an exception to Henri Lefebvre’s contention that the metaphorics of “space” in philosophy are mappings of our social experiences of producing and sharing spaces (personal, social, architectural and so on: the zones of the body and the agora). Point-set topology ontologises space: it gives a way of thinking space outside of the intuitions supplied by bodily proprioception and social practice.