$\begingroup$

What about finite topological spaces? (A useful source of stuff on these is: Algebraic Topology of Finite Topological Spaces and Applications by Jonathan Barmak.) That area studies non-Hausdorff spaces most of the time and has strong links with CW-complexes via face posets but also via the link with posets has external contacts to combinatorics and to some of your other islands.

In another direction the use of topological spaces in Logic and Theoretical Computer Science should fit somewhere. One entry point is `Topology via Logic' by Steve Vickers. This fits near to some of your existing islands so will be linked to them by bridges (probably with tolls!). There is also a use of topological spaces within Modal Logic which again looks to be distinct to the others but linked.

Finally 'pathological' is not really definable except as meaning 'outside my current interests'! Pathology is in the eye of the beholder. Spaces such as compact Haudorff spaces have a decent algebraic topology if one uses strong shape theory. This approximates these spaces by CW spaces and transfers the well loved homotopy theory of those across using procategorical methods. Even general closed subsets of $\mathbb{R}^n$ which can look pathological can be explored. There are connections between their $C^*$-algebras and their strong shape, so linking the Banach space approaches with an extended CW-approach.

(I will stop there as that leads off into non-commutative spaces, and lots of other lovely areas, such as sheaves and toposes, but is getting to the limits of stuff I know at all well!)