Continuous Maps
Definition of Continuity
Equivalent Characterizations
Properties
Homeomorphisms
Homeomorphic spaces are topologically indistinguishable.
Examples
Example 1: Identity map is always a homeomorphism.
Example 2: is homeomorphic to via .
Example 3: is not homeomorphic to (fundamental group differs).
Continuity at a Point
Closed Maps
Open Maps
Embeddings
Quotient Maps
Pasting Lemma
Extension Theorems
Tietze Extension Theorem: In a normal space, any continuous function from a closed subset to extends to the whole space.