\magnification 2200 \parindent 0pt \parskip 10pt \pageno=34 \hoffset -0.2truein \hsize 7.1truein \vsize 9.7truein \def\u{\vskip -10pt} \def\s{\vskip -5pt} \def\v{\vskip -9pt} 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 $f(B) = 1$. \end