This is an interesting question. I am pretty knowledgeable in point set topology, and I have recently read a lot of Alexander Dugin, he is very a controversial russian genius, but i find his work fascinating and he addresses your question in one of his "Noomachia" books dedicated to Ancient greek thinking (i think it's like volume 16 or something).
There are 3 epistemologies / theories of knowledge. The first one is Appollonic (in an almost Nietzschiean sense, but not quite) which is exemplified in Plato and Christianity. This is the worldview for which only the eternal / the idea / the form / the logos / the god is real, and everything else is an illusory shadow. This knowledge is modelled by Pythagorean number theory and sacred geometry (not geometry as measuring land surveying, but rather as Platonic solids and Euclidean axioms).
On the other extreme there is the logos of Cybele, which is the great subterranean mother. Out of this we get post-modernism, atomic physics and mathematically this knowledge is best modelled by chaos theory, fractals and the bizarre findings of Benois Mandelbrot.
In between the light Appollonic heaven and the dark Cybelene hell there is the fascinating domain of Dyonisis. This is essentially our phenomenal world, it is the world as a phenomenal field / horizon. Aristotle was the quintessential Dionisian thinker in Ancient Greece, and in modern times his doctrines were enriched and revived in phenomenology, Brentano began his intellectual career as a re-interpreter of Aristotle.
Topology is mathematical phenomenology. It is math of the phenomenal field. Open sets are the building blocks of topological spaces, and they are intentional acts. An open set is sort of between the finite and the infinite, it has 2 horizons, which it never reaches. Likewise, intentional acts have 2 horizons, one finite, and one infinite, they can never give you access to the transcendental, and they can not be reduced to mere finite physicality and biology.
Topology = mathematical model of the here and now, between the spiritual eternal realm above and the physical realm of finite matter. Topology is mathematics that exists between platonism and materialism.