Geometry & Topology 2024, 33: Topology

I'll obviously skip a lot of the topology basics in this chapter, because I've already read a bunch of it, and documented the contents of those chapters somewhere on here as well. I'll focus on the connections to geometry that were not as central to those books instead, so perhaps this entry might be a lot slimmer than the chapter is in the book.

As expressed through the Erlangen program, topology is a study of properties invariant under the transformation group of a topological space X. Topology infamously turns a lot of regular geometry into soup, so very few geometrical properties survive the equivalency relation of homeomorphism. We already know from homology, that separation will survive. Ordered-ness survives as well by immediate value theorem. Taken a subset V of ℝⁿ with Euclidean metric, V is closed and bounded exactly when V is sequentially compact. Sequential compactness sees every sequence contain a convergent subsequence. For any metric space, a sequentially compact subset is also compact. We recall that compactness is conserved by surjective mapping.

The topology of n-dimensional projective space is the quotient topology of ℝⁿ⁺¹\{0}. The n-sphere meets every line of ℝⁿ⁺¹ that passes through 0 in a pair of antipodal points. The topology of projective space coincides with the quotient topology of Sⁿ/±, where ± associates antipodal points. The n-Sphere is closed bounded and compact, which makes projective space compact as well by being an image of a compact space. In the case of n = 2, this topology constructs a Mobius strip.

We recall the notion of a topological base as a family of subsets satisfying finite intersections, involving every point, and containing the empty set. A topological space X with a subset topology involving the subset Y, relates closedness and compactness via the Hausdorff property. The closedness property is stronger than compactness, and compactness requires Hausdorff to satisfy closedness. If X is compact and Hausdorff, then Y is compact iff it's closed. By the relation of homeomorphisms on topological spaces, homeomorphisms are bijective, continuous and closed. This is of course known, but interestingly, if coming from the geometry side of things, this is not necessary to know up to the point, where things need to be related to homeomorphism as a property, and later when checking for the winding number. In short: It first becomes relevant when defining the fundamental group.