Topology MOC
Topology is the study of the defining features of a space. What makes two spaces the same? Topology notation in these notes
Fundamentals
The object of interest is the Topological space, which is a space together with a way of defining which subsets are open called the topology.
- Coarseness and fineness of topologies on the same space
- Topological basis, Topological subbasis
Open sets allow the definition of a Neighbourhood of a point.
Morphisms
The morphisms of interest are continuous maps. Isomorphisms are then homeomorphisms, which preserve open sets in both directions, and preserve every topological property. Other properties maps can have
Special kinds of maps
Topological properties and axiomatic topology
A Topological property is a property which is shared by any two homeomorphic spaces.
Special kinds of spaces
Internally
Sets
Sequences
Externally
I follow the structure given in Topology: A categorical approach, where we begin with the explicit topological definition, followed by a definition based on continuous maps, and finally the universal property.
- Subspace topology, Quotient topology
- Product topology (product), Coproduct topology (coproduct)
Specific topologies
- Special
- Common
- Shapes