\magnification 2000
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\R{{\bf R}}
\def\Z{{\bf Z}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\ep{\epsilon}
\def\f{\vskip 8 pt}
\def\u{\vskip -8pt}
\def\emph{{\it}}
\def\textit{{\it}}
\def\h{\vskip 5pt \hrule}
\def\b{{\hfil \break}}

$X$ is {\it locally Euclidean} if for all $x \in X$, there exists $U$ open such that $x \in U$, and there exists a homeomorphism $f: U \rightarrow f(U) \subset \R^m$ where $f(U)$ is open in $\R^m$.

Ex:  $(0, 1)$ is locally Euclidean, but $[0, 1]$ is NOT  locally Euclidean.

$X$ is an {\it $m-$manifold} if \hfil \break
(1) $X$ is locally Euclidean \hfil \break
(2) $X$ is $T_2$ \hfil \break
(3) $X$ 2nd countable.

A 1-manifold is a {\it curve}  (ex:  the circle $S^1$)

A 2-manifold is a {\it surface}  (ex:  the torus $T^2$, the projective plane $\R P^2$, the Klein bottle, $\R P^2 \# \R P^2$).

\h

The {\it support} of $\phi: X \rightarrow \R = \overline{\phi^{-1}( \R - \{0\})}$.

I.e., $x \not \in$ support $\phi$ iff there exists $U$ open such that $x \in U$ and $\phi(U) = \{0\}$.

Ex:  

\vskip 1in


Let $\{U_1, ..., U_n\}$ be a finite indexed open cover of $X$. 

An indexed family of continuous functions $$\phi_i: X \rightarrow [0, 1]$$ is a {\it partition of unity dominated by $\{U_1, ..., U_n\}$ } if \hfil \break
1)  support $\phi_i \subset U_i$ for all $i$. \hfil \break
2)  $\Sigma_{i=1}^n \phi_i(x) = 1$ for all $x$.


Ex:  $\phi_i:  \R \rightarrow [0, 1], \phi_i(x) = {1 \over 2}$ is a partition of unity dominated by $U_i = \R$, $i = 1, 2$


Note:  partition of unity for an arbitrary open cover will be defined in section 41 (one more condition, which finite covers automatically satisfy, will be needed). 
\vfill
\eject

Section 39:

A collection ${\cal A}$
of subsets of $X$ is {\it locally finite} if for all $x \in X$, there exists $U$ open such that $x \in U$ and $U$ intersects only finitely many elements of ${\cal A}$

Ex:  ${\cal A} = \{(n, n+ 2) ~|~ n \in \Z  \}$ is locally finite.

\vfil
Ex:  ${\cal C} = \{(n, n+ 2) ~|~ n \in \Z_+  \}$ is locally finite.
\vfil

Ex:  ${\cal D} = \{(0, n) ~|~ n \in \Z_+  \}$ is NOT locally finite.
	\vfil

Ex:  A finite collection of sets is locally finite.

The indexed family $\{A_\alpha ~|~ \alpha \in J \}$ is a {\it locally 
finite indexed family} in $X$ if for all $x \in X$, there exists $U$ open such that $x \in U$ and $U$ intersects $A_\alpha$ for only finitely many $\alpha$.

Ex:  If $A_i = \R$ for all $i \in \Z$, then $\{A_i ~|~ i \in \Z \}$ is NOT  a  locally finite indexed family in $X$, but $\{A_i ~|~ i \in \Z \}$, as a collection of set(s), is locally finite (since it contains only one set).
\vfil

\eject

A collection ${\cal A}$
of subsets of $X$ is {\it countably locally finite} if ${\cal A}$ can be written as a countable union of collections ${\cal A}_n$, each of which is locally finite.

 Ex:  ${\cal D} = \{(-n, n) ~|~ n \in \Z_+  \}$ is countable locally finite.  Let  
${\cal D}_k  = \{(-n, n) ~|~ n \in [k, k+2]  \}$, $k \in 2\Z_+$. 

Note ${\cal D} = \cup_{k \in 2\Z_+} {\cal D}_k  $ and 
${\cal D}_k $ is locally finite since it's finite. 

Let  ${\cal A}$ be a collection of subsets of $X$.   A collection ${\cal B}$
of subsets of $X$ is a {\it refinement} of ${\cal A}$  if for all $B \in {\cal B}$, there exists $A \in {\cal B}$ such that $B \subset A$.  

If the elements of ${\cal B}$ are open then ${\cal B}$ is an {\it open refinement} of ${\cal A}$.
\b
If the elements of ${\cal B}$ are closed then ${\cal B}$ is a {\it closed refinement} of ${\cal A}$.



A simply ordered set $X$ is {\it well ordered} if every nonempty subset of 
$X$ has a smallest element (ie $A \subset X$, $A \not= \emptyset$ implies 
$min(A)$ exists and $min(A) \in A$).

Ex:  $\Z$ is NOT well-ordered.
\b
Ex:   $\Z_+$ is well-ordered
\b
Ex:  $\R_+$ is NOT well-ordered.

The Well-ordering theorem:  If $X$ is a set, there exists an order relation on $X$ that is  well-ordered.

\end