Section 35*: The Tietze Extension Theorem

1. Tietze Extension Theorem: $X$ is normal, $A$ is closed in $X$ , $f:A\to B$ where $B=[0,1]$ or $B=\mathbb{R}$ is continuous, then $f$ can be extended to a continuous function $X\to B$ .
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 $Y$ has the universal extension property if every function that maps continuously a closed subset of a normal space into $Y$ can be extended onto the whole space.
8. If $Y$ is normal then it has the universal extension property iff it is an absolute retract.
9. A space homeomorphic to a retract of $\mathbb{R}^J$ has the universal extension property.
10. The adjunction space $Z$ for spaces $X$ , $Y$ and a continuous function $f:A\to Y$ where $A\subseteq X$ , is the quotient space of the space $X\cup Y$ obtained by identifying a point $y$ with all points in $f^{-1}(y)$ . Let $p$ be the quotient map.
11. If $A$ is closed, $Z$ is Hausdorff.
12. $Y$ is homeomorphic to $p(Y)$ , and if $A$ is closed then $p|Y$ is a closed imbedding.
13. $X-A$ is homeomorphic to $p(X-A)$ , and if $A$ is closed then $p|(X-A)$ is an open imbedding.
14. If $A$ is closed, $X$ and $Y$ are normal, then the adjunction space is normal.
15. The coherent topology: given a sequence of spaces $X_i\subseteq X_{i+1}$ such that $X_i$ is closed in $X_{i+1}$ we define the topology on their union $X$ coherent with the subspaces $X_n$ : $U$ is open in $X$ iff $U\cap X_n$ is open in $X_n$ for every $n$ .
16. $X_i$ is a closed subspace of $X$ .
17. If $f|X_i$ is continuous for every $i$ then $f$ is continuous.
18. If every $X_i$ is normal then $X$ is normal.