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26. Compact Sets (continued)

Defn:  A collection {\cal C} is said to have the {\bf finite intersection
property} if for every finite subcollection \break
$\{C_1, ..., C_n\} \subset
{\cal C}$, $\cup_{i = 1}^n C_i \not= \emptyset$.

Example 1: $\{(-n, n) ~|~ n = 1, 2, 3, ... \}$ has/does not have finite 
intersection
property.

Example 2:  $\{(n, n + 2) ~|~ n \in {\cal Z} \}$ has/does not have finite 
intersection
property.

Example 3:  $\{(0, {1 \over n}) ~|~ n = 1, 2, 3, ... \}$ has/does not have
finite intersection
property.

Thm 26.9:  $X$ is compact if and only if for every collection ${\cal C}$ of closed sets in $X$ having the finite intersection property, $\cup_{C \in {\cal C}} C \not= \emptyset$.

Thm 27.3:  A subspace $A$ of $R^n$ (with standard topology) is compact if and only if it is closed and bounded in the euclidean or square metric.

30. Countability Axioms

Defn:  $X$ is said to have a {\bf countable basis at the point $x$} if
there exists a countable collection ${\cal B} = \{B_n ~|~ n \in Z_+\}$ of
neighborhoods of $x$ such that if $x \in U^{open}$ implies there exists a
$B_i \in {\cal B}$ such that $B_i \subset U$.

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

Defn:  A space is second countable if it has a countable basis.

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

31. Separation Axioms

Defn:  $X$ is {\bf regular} if one-point sets are closed in $X$ and if for all closed sets $B$ and for all points $x \not\in B$, there exist disjoint open sets, U, V, such that $x \in U$ and $B \subset V$.


Defn:  $X$ is {\bf normal} if one-point sets are closed in $X$ and if for all pairs of disjoint closed sets $A, ~B$, there exist disjoint open sets, U, V, such that $A \subset U$ and $B \subset V$.

Normal implies regular implies Hausdorff implies $T_1$.

Thm 32.3:  Every compact Hausdorff space is normal.


HW (choose 3 - 4):  p. 170:  1, 2, 3, 4, 5,  p. 199: 8*

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