Connectedness
Central Question
How can open sets express that a space cannot be split into separate pieces?
Why This Matters
Connectedness provides an early topological invariant and reveals the structural reason continuous real-valued functions take intermediate values.
Learning Prompt
Motivate connectedness through the problem of distinguishing spaces. Derive its definition using separation into open sets. Connect connected subsets of to intervals and the Intermediate Value Theorem. Use connectedness as a topological invariant.