Separation Properties and Constructing Continuous Functions

1. Hausdorff: $\forall x, y \in X$, $x \neq y$, there exists $A, B$ open such that $x \in A, y \in B$, $A \cap B = \emptyset$.
2. Normal: $\forall E, F \subset X$, closed, $E \cap F = \emptyset$. Then, there are $A, B$ open such that $E \subset A$, $F \subset B$, $A \cap B = \emptyset$.
   

   Examples:
   

   1. Metric spaces are normal
   2. Compact Hausdorff spaces are normal
   Urysohn's Lemma
   Let $X$ be normal, $A, B$ closed, and $A \cap B = \emptyset$. Then, there exists an $f \in C(X)$ such that $f\vert _A = 1$, $f\vert _B = 0$.
   

   Tietze's Extension Theorem
   Let $X$ be normal, $F$ closed. Given $f \in C(F)$, there is $g \in C(X)$ such that $g\vert _F = f$ and $g$ is bounded.

   Locally compact: For all $x \in X$, there exists $C$ compact, $U$ open such that $x \in U \subset C$.

   One Point Compactification: Define $\tilde X$ to be $X \cup \{ \infty \}$ and assume $\infty \notin X$. Given $\tilde X$ a topology consisting of all open sets in $X$ and $C^c \cup \{ \infty \}$ for all $C \subset X$, compact. Then $\tilde X$ is a compact Hausdorff space, and hence normal.
   

   Wednesday, February 1, 2006:
   

   Theorem 1.8.1   Tietze's Extension Theorem (for locally compact Hausdorff spaces.
   Let $X$ be such a space and $F \subset X$ compact. Given $f:F \rightarrow \mathbb{R}$, continuous, there is $g:X \rightarrow \mathbb{R}$, continuous such that $g\vert _F = f$.
   Moreover, we may arrange that the support of $g$ is compact.