What are the different ways that topologies have been presented or defined over time.
Topology is about a structure of cohesion on the points of a space. The question of what constitutes cohesion has been asked since earliest times, essentially arising from the paradoxes of Zeno. Aristotle thought of the physical continuum in terms of potentiality and actuality.
The same question was tackled by Hegel who thought of a moving point as being here and there at the same time. This is worth considering since an open set in pointless topology, is here and also over there. I mean by that, it is not localised to a point. The question of what constitutes the physical continuum was tackled in Weyls Space, Time & Matter, first published in 1918.
This sense of a 'motion' culminating in a 'limit' is axiomatised via nets and which generalises the definition of a limit of a sequence. This is exactly equivalent to the traditional notion of a topology by open or closed sets and deserves to be better known that it is now, especially given that limits are usually introduced at school and hence, pedagogically speaking, generalised limits - aka nets - are a natural way to introduce topology.
We have the classical notion of a topology in terms of nets (or filters), opens (or closeds) and interior (or closure operators). There are more general notion of topologies such as pseudo-topologies and convergence spaces. They aim to have better categorical properties. For example, Top, the category of topological spaces does not have exponentials, that is a natural topology on mapping spaces. Whereas Conv, the category of convergence spaces, being a quasi-topos, does; and moreover, Top embeds in it as a full subcategory. This means Conv extends the notion of topology.
But there is more to the question of cohesion than simpe topology. We can also ask how smoothly do they cohere. This of course is differentiability and this has also been axiomatised. For example, there is diffeology, which was put forward by Souriau in the 1980s, and which aims to do for smoothness what topology does for continuity. There is also the notion of bornology, which does the same for boundedness - in fact, one mathematician has said that in functional analysis, bornology was more natural than topology. In fact, there is also the notion of convexology , a neologism for a similar notion for convex structures. This might seem to be far from the notion of cohesion but it's possible that the fine structure of a space may be convex as in locally convex spaces and where the convexity is crucial to define a well-behaved notion of differentiability and hence of smoothness.
It's worth adding that categories can be topologised. Here, what is used is an abstraction of the notion of covering from topology and what is called a Grothendieck topology. This was formulated by Grothendieck in the early 1960s.
It's probably also worth saying that uniform continuity had a separate axiomatic development with the development of uniform spaces and uniformities. And also that completeness, which is about the fine structure of points; in particular, that there are no 'missing' points and hence 'complete' also has a separate axiomatix treatment as Cauchy spaces. The name here obviously derives from Cauchy sequences or nets which are traditionally used to define the completeness of a metric space, or more generally, of a metrisable topological space.