Topology

Checkpoints

• Accumulation point Basis Defines accumulation points (limit points) of a set in a topological space — points that every open neighbourhood of which touches the set at some point other than itself — and examines how topology determines which points qualify.
• Boundary (Topology) Basis Defines the boundary of a set in a topological space as the points belonging to the closure of both the set and its complement, establishes the canonical three-way partition of the space, and explores how open, closed, and clopen sets are characterised by their boundaries.
• Closure (Topology) Basis Defines the closure of a set in a topological space as the smallest closed set containing it, establishes its equivalent characterisations via derived sets and intersections of closed sets, and works through its key properties.
• Derived Set Basis Introduces the derived set of a subset of a topological space — the collection of all its accumulation points — and shows how it characterises closedness and forms the bridge to the closure operator.
• Interior (Topology) Basis Defines the interior of a set in a topological space as the largest open subset it contains, and explores how interior points capture what it means to be 'strictly inside' a set.
• Topology Space Basis Introduces topological spaces — a generalisation of metric spaces that axiomatises the notion of 'open set', freeing the theory of closeness from any specific distance function.