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Thm 27.1:  Let $X$ be a simply ordered set having the least upper bound property.  In the order topology, each closed interval in $X$ is compact.

Cor 27.2:  Every closed interval in $R$ is compact.

Thm 27.3:  A subspace $A$ of $R^n$ is compact if and only if it is closed an bounded in the Euclidean metric or the square metric.

Thm 27.4 (extreme value theorem):  \hfil \break
$f^{cont}: X^{compact} \rightarrow
Y^{order ~top}$.  Then there exists points $c, d \in X$ such that $f(c )
\leq
f(x) \leq f(d)$ for every $x \in X$.

Thm 27.6 (Uniform continuity theorem).   \hfil \break
$f^{cont}: X^{compact ~metric}
\rightarrow Y^{metric}$, then $f$ is uniformly continuous.

Defn:   $X$ is {\bf limit point compact} if every infinite subset of $X$ has a limit point in $X$.

Thm 28.1:  Compactness implies limit point compactness, but not conversely.

Defn:   $X$ is {\bf sequentially compact} if every sequence in $X$ has a subsequence which converges in $X$.

Thm 28.3:  If $X$ is metrizable, compact, limit point compact, and
sequentially compact are equivalent.

(29) Defn:  If $\overline{X} = Y$ is compact Hausdorff, then $Y$ is a {\bf
compactification } of $X$.  If $Y - X$ = one point, then $Y$ is the
{\bf one-point compactification} of $X$

30:  The countability axioms

Defn:  $X$ is first countable if  $X$ has a countable basis at each of its points.

Ex: $R_{l}$ is first countable:  $\{[x, r) ~|~ r > x, r \in Q \}$

Defn:  $X$ is second countable if $X$ has a countable basis.

Ex:  $R_{l}$ is NOT second countable.

Ex:  $R$ is second countable:  $\{(s, r) ~|~ s < r, s, r \in Q \}$

Defn:  $A$ is dense in $X$ if $\overline{A} = X$

Defn:  $X$ is separable if $X$ has a countable dense subset.

Ex:  $Q$ is dense in $R$  ($\overline{Q} = R$) and hence $R$ is separable.


31:  The Separation Axioms:  

Defn:  $X$ is $T_0$ if  for all $x_1, x_2 \in X$ such that $x_1 
\not= x_2$, there exists a neighborhood containing one of the points, but
not the other one.  

Ex:  $X = \{a, b, c\}$, ${\cal T} = $ indiscrete topology is NOT $T_0$.

Ex:  $X = \{a, b, c\}$, ${\cal T} = \{ \emptyset, \{a\}, \{a, b\},\{a,
b, c\}\}$ is $T_0$.

Defn:  $X$ is $T_1$ if every one point set is closed.

Lemma:  $X$ is $T_1$ if and only if for all $x_1, x_2 \in X$ such that $x_1
\not= x_2$, there exists neighborhoods $U_1$ and $U_2$ of $x_1$ and $x_2$,
respectively, such that $x_1 \not\in U_2$ and $x_2 \not\in U_1$.

Ex:  $X$ with the co-finite topology is $T_1$.

Defn:  $X$ is Hausdorff space ($T_2$) if for all $x_1, x_2 \in X$ such that
$x_1 \not= x_2$, there exists neighborhoods $U_1$ and $U_2$ of $x_1$ and
$x_2$, respectively, such that $U_1 \cap U_2 = \emptyset$.

Thm 17.8:  Every finite point set in a Hausdorff space $X$ is closed.

Ex:  Let $S = \{(a, b) ~|~ a < b, a, b, \in R\} \cup \{r - A\}$ where $A =
\{{1 \over n}~|~ n = 1, 2, 3, ...\}$.  Then the topology on $R$ generated
by the subbasis $S$ is finer than the usual topology is is thus, $T_2$.

Defn:  $X$ is {\bf regular} ($T_3$) if $X$ is $T_1$ and if for every pair
consisting of a point $x$ and a closed set $B$ such that $x \not\in B$,
there exists neighborhoods $U_1$ and $U_2$ such that $x\in U_1$ and $B
\subset U_2$ and $U_1 \cap U_2 = \emptyset$.



Defn:  $X$ is {\bf normal} ($T_4$) if $X$ is $T_1$ and if for every pair of disjoint closed sets $A, B$,
there exists neighborhoods $U_1$ and $U_2$ such that $A \subset U_1$ and $B \subset U_2$ and  $U_1 \cap U_2 = \emptyset$.

\s		
Ex: Let $R$ have the topology ${\cal T}' = \{(a, +\infty) ~|~ a \in R \}$.
Then $R$ is NOT normal with this topology because it is not $T_1$, but it
(vacuously) satisfies the second condition of normality.


Thm 32.3:  Every compact $T_2$ space is $T_4$.
\s
Note that we have basically proved that every compact $T_2$ space is $T_3$ (lemma 26.4).


Defn $X$ is completely regular ($T_{3.5}$)if $X$ is $T_1$ and if 
every pair 
consisting of a point $a$ and a closed set $B$ such that $a \not\in B$,
there exists a continuous function $f: X \rightarrow [0, 1]$ such that
$f(a) = 0$ and $f(B) = 1$.

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