I'm an undergraduate math student. I've taken several courses in topology yet I always feel like I'm at the most basic level of knowledge there is (I think of it as a natural number - it can be big and get bigger as much as I want, but it will never even grasp infinity). Introductory books such as Willard's, Dugundji's, Engelking's and Munkres' really are veeery elementary, but I can't seem to find something more advanced anywhere.
There is differential topology, continuum theory, knot theory, algebraic topology (and all the abstraction that become the theories of homology and homotopy) and the theory of topological vector spaces. I've come across some concepts like door spaces, fuzzy topological spaces, hyperspaces, topological data analysis and persistent homology, the connectification of spaces (analogous to compactifications) and even the strange concept of the "tight span" of a metric space. We have the newborn hyperspace of non-trivial sequences, and I've also read about toposes and the categorical notion of a closure operator.
There are also websites like Dan Ma's Topology Blog, $\pi$ Base (which is a web-version of Counterexamples in Topology) and this strange site with a ridiculous amount of examples in continuum theory.
It seems like a vast world of topology is just developing and evolving, but it seems quite inaccesible. I'd like to learn a lot more than the very few things I mentioned, because I know there is a lot more out there, however I simply don't know how to do it.
How do I start? Where do I go next?
I think at the point you are topology splits into various distinct fields that have little to with each other and can have very different prerequisites. Some big areas ordered by how topological I think they are: