Section 35*: Problem 1 Solution »

Section 35*: The Tietze Extension Theorem

  1. Tietze Extension Theorem: is normal, is closed in , where or is continuous, then can be extended to a continuous function .
  2. The idea: if the range of a function is [-r,r] using the Urysohn lemma construct a continuous function such that its range is [-r/3,r/3] and it is never more than 2r/3 from the original function, then take the difference and do it again.
  3. A retract is a subspace of a topological space such that there exists a surjective retraction onto it (see the Supplementary exercises of Chapter 2).
  4. A retraction is a quotient map.
  5. A retract of a Hausdorff space is a closed subspace.
  6. A normal space is an absolute retract if whenever a closed subspace of a normal space is homeomorphic to it, the subspace is a retract.
  7. A space has the universal extension property if every function that maps continuously a closed subset of a normal space into can be extended onto the whole space.
  8. If is normal then it has the universal extension property iff it is an absolute retract.
  9. A space homeomorphic to a retract of has the universal extension property.
  10. The adjunction space for spaces , and a continuous function where , is the quotient space of the space obtained by identifying a point with all points in . Let be the quotient map.
  11. If is closed, is Hausdorff.
  12. is homeomorphic to , and if is closed then  is a closed imbedding.
  13. is homeomorphic to , and if is closed then is an open imbedding.
  14. If is closed, and are normal, then the adjunction space is normal.
  15. The coherent topology: given a sequence of spaces such that is closed in we define the topology on their union coherent with the subspaces : is open in iff is open in for every .
  16. is a closed subspace of .
  17. If is continuous for every then is continuous.
  18. If every is normal then is normal.