Introduction

• Algebraic topology is the interplay between continuous and discrete mathematics, for topological/geometrical continuum, A.T allows their realizations to be of an algebraic nature.
• Classical A.T consists of construction and use functors to study/extract-information from topological category. Two important views are homology and homotopy. These implies the category of topological spaces has itself a kind of hidden algebraic structure and can been manifested only in techniques of A.T, these properties can not been obtained using simply point-set topology.

Topological space

This chapter serve as a set up of some point-set topological notions, the most important notions are quotient space(quotient set with quotient topology) and transformation groups.

Basic notions

• Topological space is defined axiomatically with open sets, dually closed sets. Related notions including: closure(also defining dense subset), interior, boundary, etc. There’s notion of topological basis, also a weaker one known subbasis. There’s also subspaces, with subspace/induced topology defined on subset.
• There are many topological structure can be defined. We then have the notion of coarser/finer topology. The morphisms are continuous maps defined via open sets, the morphisms are usually refered to as open/close morphism. The isomorphism in TOP is homeomorphism, we have to make sure the inverse is also continous(a morphism).
• The notion of (open) neighbourhood is usually used, it is used to define continuity at some point. This notion is more useful in metric spaces, with topology defined respect to the metric, using open neighbourhoods. This type of topology is more related to analysis.
• There are some properties we can attach to a topological space. Separation axioms have four types: Hausdorff(separated), (complete) regular, normal. Another one is (local) compactness(defined using open coverings).
• Associated with compactness is the technique of compactification, most used one is known as one-point compactification. The proper morphism is associated with compactness.
• There’s theorem implying locally compact set in Hausdorff space is locally closed. Locally closed set in locally compact space is locally compact.

There are many theorems related to openness, Hausdorffness, compactness, etc in the literature, they seem not that important to me.

• Associated with morphisms we have notions of sections, in topological context they are assumed to be continuous. Same for rectraction.

Categorical constructions

• Product of topological spaces is defined with product topology on product of sets. The product topology is the coarsest topology making the projections continuous. The universal properties in this book is frequently framed as “the unique(by construction from universal properties in SET) map is continuous if the associated ones are”. This is just checking the definition is indeed a morphism, this is important and we should also check well-defineness.
• Dually we have coproduct which is refered to as topological sum. The subspace topology of components is the original ones, this means they are open in the sum.
• Finer constructions are fiber product and fiber coproduct, the former is refered to as pull back defined with subspace topology, the latter is refered to as push out defined by quotient topology via coproduct morphism.
• There’s notion of embedding, this is common in describing the topological object in terms of solution set of equations, they are defined with subspace topology. The embedding is of categorical nature
• The quotient space is important, it express the idea of pasting points by identifying them with equivalence relations usually through morphisms. Specificly the quotient topology is defined via surjective map, equivalently canonical projections to quotient sets. This morphism is known as identification or quotient map.
• Associted with quotient is the idea of constructing larger spaces by pasting smaller pieces. These process will finally need quotient, moding out some equivalence relation, this quotient process is intuitively identifying some structures in common, thus pasting different pieces together. Specific example include adjunction space, which is also a special case of push out of inclusion.
• More generally, we have the notion of clutching, the definition is rather subtle, involving labeled subsets and bijections. The bijections define the notion of pasting, the datum lead to equivalence relations on disjiont sum(largest set)between points comes from different pieces. By quotienting we have well defined spaces and injections identifying different pieces. The bijections serve for transition between different pieces pasting them together.

Topological groups

• A topological group is a group with topological structure defined on set making the inverse and multiplication continous. The left action of group on itself is known as left transition and is a homeomorphism. Examples include matrice groups some of them are identified with topological spaces like n-sphere, n-torus, etc. Notions like subspace and quotient can be defined.
• Group action of topological group on topological spaces define category G-TOP, consisting of G-spaces(like G-module, G-set) and equivariant morphisms(G-map). There’s related notions like orbit space by quotienting out the G action, stabilizer fixed point set, etc.
• The equivariant continuous maps between topological spaces induce natrually continuous maps between corresponding G-spaces.
• Equivariant homotopy or G-homotopy is defined as a homotopy such that at all time the morphism is a G-map.
• The quotient space is defined on quotient set, the quotient space may not be a quotient group. The quotient space is defined with natural G-action, it is refered to as homogeneous space. It’s important in geometry.
• By orbit-stabilizer theorem, the homogeneous space constructed by mod out the some stabilizer is identified with the orbit(transitive) we usually refer the latter. Important examples include sphere, projective spaces, Grassmann manifolds, Stiefel manifolds.
• The homogeneous G-spaces with the mod out subgroup closed in G together with G-maps form orbit category.
• The action on real/complex vector space is known as (G-modules) real/complex representation if the translations are linear maps.
• Given a right G-space and a left G-space we can construct product with a left G action. Then by quotienting the action we obtain a orbit space. This constuction is important.

The fundamental group