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26.  Compact Spaces

Defn:  A collection ${\cal C}$ of subsets of $X$ is said to have the
{\bf finite intersection property} if for every finite subcollection
$\{C_1, ..., C_n\}$ of ${\cal C}$, $C_1 \cup ...\cup C_n \not=
\emptyset$

\end
Thm 4:  If $f:D \rightarrow R$ is continuous and if $D$ is compact,
then $M = sup_{\{p \in X\}} f(p)$ and $m = inf_{\{p \in X\}} f(p)$
exist and there exist points in $p, q \in D$ such that $f(p) = M$ and
$f(q) = m$.


Defn:  $f: D
\rightarrow Y$.
is {\bf uniformly continuous} if
all $\epsilon > 0$, there exists a $\delta > 0$ such that

\centerline{
$d_X(x, p) < \delta$ implies $d_Y(f(x), f(p)) < \epsilon$}
\vskip -10pt
 for all $x \in
D$.

Defn:  If $f:D \rightarrow Y$ is continuous and if $D$ is compact,
then $f$ is uniformly continuous.


\end

Thm 3.12 (Heine-Borel Thm)  Suppose that a family ${\cal F}$ of open
intervals covers the closed interval $I = [a, b]$.  Then a finite
subfamily of ${\cal F}$ covers $I$.


A family of sets, ${\cal F} = \{F_\alpha ~|~ \alpha \in A\}$, covers
the
set $S$ if 

\centerline{$S \subset \cup_{\alpha \in A}F_\alpha$}

${\cal F}$  is said to be a cover of $S$.  


Defn:  In a metric space $X$, a set $A$ is compact if
every sequence contains a
subsequence which
converges to a point in $A$.

A set $A$ is bounded if there exists an open ball, $B(p, r)$ such
that $A \subset B(p, r)$.

Thm 6.13:  Compact metric implies closed \& bounded.

Thm 6.14:  In ${\cal R}^n$ with standard metric, 
{closed \& bounded implies
compact.}

Thm: Convergence implies Cauchy.

Thm:  In a compact metric space, Cauchy implies convergence.

Thm:  In $R^n$, Cauchy implies convergence (even though $R^n$ is not
compact).



Thm 6.15 (Heine-Borel Theorem)
\v
Suppose $X$ is a metric space and $A \subset X$, then the following
are
equivalent:
\v
(1.)  Every sequence contains a
subsequence which
converges to a point in $A$.
\v
(2.)  Every open cover of $A$ contains a finite subcover.

\end