Peirce was one of the first to invoke topology in his metaphysics. The notion of continuum and continuity is central to his account of the basic philosophical categories, and he was even led to study three-valued logic by topological considerations (status of points on the boundaries between regions). See Peirce’s Topological Concepts by Havenel:
"Peirce thinks that the study of topology could help to
solve many philosophical questions. For example, the idea of continuity is of prime importance for Peirce’s synechism, and topology is “what the philosopher must study who seeks to learn anything about continuity from geometry” (NEM 3.105). In Peirce’s reasonings concerning the nature of time and space, topological concepts are essentials. Moreover, in his logic of existential graphs and in his cosmology, the influence of topological concepts is very apparent."
Introduction: The Becoming Topological of Culture by Lury et al. is a recent survey of analogical application of topological ideas to cultural theory:
"The influence of topology in social and cultural theory in recent decades is immense. Topological ideas have been a significant source of
inspiration across many social science disciplines, including philosophy,
sociology, political science, psychology, anthropology, geography and
economics. Topological ideas have fed into and transformed multiple
specific fields of study. So, for example, as Marres (2012) observes, in
the field of social studies of technology, topological ideas have helped
dismantle the view that technology and society occupy different domains,
contributing instead to the concept of a heterogeneous ‘assemblage’,
which is heterogeneously composed of social, technical and natural enti
ties (Latour, 1987)...
In all these approaches it is relationality that is important, and topological concepts have enabled social theory to move beyond a reliance on the mechanical and organic to include transductive and transitive modes of relating (Simondon, 1992), and helped turn space and time from ‘a priori’ into ‘a posteriori’ categories (Lash, 2009). But what is proposed in this introduction is not simply the transposition of topological ideas onto the field of culture. Instead we are interested in teasing out an epochal transformation in the intersection between the form and content of cultural expression... our proposal is that topology is now emergent in the practices of ordering, modelling, networking and mapping that co-constitute culture, technology and science.".
For a critical philosophical discussion of this "topological turn", anticipated by Lacan and Badiou, see On Topology by Phillips, who is advocating a different, "non-quantitative", "inexact", conception of topological in culture, one inspired by Heidegger:
"This article questions the novelty of this ‘becoming topological of culture’ and digs into a deeper historicity in order to identify the trends that may be said to support the development of topology in the current situation... For instance, mathematical objects, despite intersections, seem relatively distinct from those of the social sciences. But the idea of a topological turn in social theory suggests at the very least that the latter, the objects of political and cultural theory, say, can be addressed as if they were of the same kind as those of the former.
[...] I think, though, that the high level of abstraction required for
the construction of a topological space might give the impression that
some kind of labile inexactitude operates there too – which I believe is
false. You don’t really need mathematics for labile inexactitude. The
capacity to absorb a vocabulary of topological transformation (mapping,
open sets, neighbourhoods, homologies and so on) into an already existing
vocabulary derived from the structuralism of the turn of the last
century might better be examined by taking the historicity of concepts
as a guide."
