From Metric Spaces to Topology
Central Question
Which parts of analysis survive when we retain open sets but discard numerical distance?
Why This Matters
Topology isolates the structure behind continuity and neighbourhoods, allowing familiar analytic ideas to work beyond metric spaces.
Learning Prompt
Start from metric spaces. Show which ideas in analysis genuinely require distance and which only require open sets. Derive why open sets are sufficient to define continuity and motivate the topology axioms.