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 Uniform Convergence of Functions Ordinary Differential Equations Totally Bounbdedness
Topological Spaces
Topological Spaces Closure, Interior and Boundary Hausdorff Space Compactness of Topological Space Continuous Functions on Topological Spaces Connectedness and Paths Nets Isometries and Homeomorphisms
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Acknowledgement
This part is mainly based on the ANU course MATH2320 in 2024 and lecture notes from Warwick course MA260.