Topology of the plane
- Functions of two variables as maps . Domains in the plane. Open and closed domains.
- Convergence of planar points: . Alternative description: any square contains almost all elements of the sequence.
- Limits of functions: , . We say that , if for any sequence of points converging to , the sequence converges to as . We say that is continuous at , if .
- Exercise: is continuous at , if the preimage , of any interval , contains the intersection for some sufficiently small .
- Exercise: if is continuous at all points of the rectangle , then for any the function , defined by the formula is continuous on . Can one exchange the role of and ? Formulate and think about the inverse statement.
- Exercise: Check that the functions and are continuous. What can be said about the continuity of the function ?
- Exercise: formulate and prove a theorem on continuity of the composite functions.
- Exercise: Give the definition of a continuous function for . Planar curves. Simple curves. Closed curves.
- Intermediate value theorem for curves. Connected sets, connected components. Jordan lemma.
- Rotation of a closed curve around a point. Continuity of the rotation number. Yet another “Intermediate value theorem” for functions of two variables.