\magnification 2400 \parindent 0pt \parskip 12pt \hsize 7.2truein \vsize 9.7 truein \hoffset -0.45truein \def\u{\vskip 0pt} \def\w{\vskip -10pt} \def\v{\vskip 0.5in} Cor 29.3: If $X$ is locally compact and if $A$ is closed in $X$, then $A$ is locally compact. Cor 29.3: If $X$ is locally compact Hausdorff and if $A$ is open in $X$, then $A$ is locally compact. Cor 29.4: $X$ is homeomorphic to an open subspace of a compact Hausdorff space if and only is $X$ is locally compact Hausdorff \hrule Thm 30.3b: 2nd countable implies separable. Idea of proof: Lemma: If $X$ is 2nd countable, $A \subset X$, and the subspace topology on $A$ is the discrete topology, then $A$ is countable. Idea of proof: \vfill Thm 30.2a. A subspace of a second countable space is second countable. A countable product of second countable spaces is second countable. \eject Thm 30.2b. A subspace of a first countable space is first countable. A countable product of first countable spaces is first countable. Metric spaces are 1st countable. $R^\omega$, product: $R^\omega$, uniform: $R^\omega$, box: Defn: $X$ is countably compact if $X \subset \cup_{n=1}^\infty U_n$, $U_n$ open implies there exists $n_1, ..., n_k$ such that $X \subset \cup_{i=1}^k U_{n_i}$. Defn: $X$ is Lindelof if $X \subset \cup_{a \in A} U_a$, $U_a$ open implies there exists $a_i \in A$ such that $X \subset \cup_{i=1}^\infty U_{a_i}$. 2nd countable does not imply countably compact. Thm 30.3a: 2nd countable implies Lindelof. \eject ${\cal R}_{\cal L}$ = ${\cal R}$ with the lower limit topology is Separable: 1st countable: Not 2nd countable: \vskip 0pt Lindelof: Step 1: Every open covering has a countable subcover iff every open covering of basis elements has a countable subcover \vfill ${\cal R}_{\cal L} \times {\cal R}_{\cal L}$ is not Lindelof. Hence the product of Lindelof spaces need not be Lindelof. HW: A subspace a of Lindelof space need not be Lindelof. Hint: The ordered square $= [0, 1] \times [0, 1]$ with the order topology using the dictionary order. \eject 31 The separation axioms. \u Defn: $X$ is $T_0$ if $x, y \in X$ $x \not= y$, then there exists an open set $U$ such that $x \in U$, $y \not\in U$ OR $y \in U$, $x \not\in U$. \u Ex: $\{(a, \infty) ~|~ a \in {\cal R}\}$ Defn: $X$ is $T_1$ if $x, y \in X$ $x \not= y$, then there exists open sets $U$, $V$ such that $x \in U$, $y \not\in U$ and $y \in V$, $x \not\in V$. \u Thm: $X$ is $T_1$ iff points are closed in $X$. \u Ex: finite complement topology Defn: $X$ is Hausdorff if $x, y \in X$ $x \not= y$, then there exists disjoint open sets $U$, $V$ such that $x \in U$ and $y \in V$. \u Ex: ${\cal R}_K$ where ${\cal R}_K$ has basis \hfil\break $\{(a, b) ~|~ a, b \in {\cal R}, a < b \} \cup \{(a, b) - K ~|~ a, b \in {\cal R}, a < b \}$ where $K = \{ {1 \over n} ~|~ n = 1, 2, 3, ... \}$ Defn: $X$ is regular if $X$ is $T_1$ and if $x \in X$, $C$ closed in $X$ and $x \not\in C$, then there exists disjoint open sets $U$, $V$ such that $x \in U$ and $C \subset V$. \u Ex: ${\cal R}_{\cal L} \times {\cal R}_{\cal L}$ Defn: $X$ is normal if $X$ is $T_1$ and if $C, D$ disjoint closed sets in $X$, then there exists disjoint open sets $U$, $V$ such that $C \subset U$ and $D \subset V$. Ex: ${\cal R}_{\cal L}$ Lemma 31.1: Let $X$ be $T_1$. \w a.) $X$ is regular iff for all $x\in X$ and for all neighborhoods $U$ of $x$, there exists neighborhood $V$ of $x$ such that $\overline{V} \subset U$. \w b.) $X$ is normal iff for all $C$ closed in $X$ and for all open set $U$ containing $C$, there exists an open set $V$ containing $C$ such that $\overline{V} \subset U$. Lemma 31.2: A subspace of a $T_i$ space is $T_i$ for $i = 0, 1, 2, 3$, but not $i = 4$ \w A product of $T_i$ spaces is $T_i$ for $i = 0, 1, 2, 3$, but not $i = 4$ 32: Normal Spaces: Thm 32.1: Every regular space with a countable basis is normal. Thm 32.2: Every metrizable space is normal. Thm 32.3: Every compact Hausdorff space is normal. Thm 32.4: If $X$ has the order topology then $X$ is normal. \end Linn.iowaassessors.com