Introduction
In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology.
The fundamental concepts in point-set topology are continuity, compactness, and connectedness:
- Continuous functions, intuitively, take nearby points to nearby points.
- Compact sets are those can be covered by finitely many sets of arbitrarily small size.
- Connected sets are sets that cannot be divided into two pieces that are far apart.
Contents
Metric Spaces
Metric Spaces
Interior, Exterior and Boundary
Limit Points and Closure
Open and Closed Sets
Sequence and Convergence
Compactness of Metric Space
Complete Metric Space
Continuity on Metric Spaces
Convergence of Functions
Totally Bounbdedness
Topological Spaces
Topological Spaces
Closure, Interior and Boundary
Separation and Hausdorff Spaces
Compactness of Topological Space
Continuous Functions on Topological Spaces
Connectedness and Paths
Nets
Isometries and Homeomorphisms
More …
Acknowledgement
This part is mainly based on the ANU course MATH2320 in 2024 and lecture notes from Warwick course MA260.