\magnification 2000
\parindent 0pt
\parskip 5pt
\pageno=31
\hoffset -0.2truein
\hsize 7.1truein
\vsize 9.9truein
\def\u{\vskip -5pt}
\def\s{\vskip -4pt}
\def\v{\vskip -5pt}


$bd E = \overline{E} \cap \overline{X - E} = \overline{E} - E^o$
\u
$p \in X$ is a boundary  point of $E$ if $p \in bd E$.
\u
$p$ is an isolated point of $E$ if $p \in E$, but $p \not\in E'$.
\u
$p \in X$ is a limit point of $E$ if 
for all $U$ open such that $x \in U$,
$U \cap E - \{x\} \not= \emptyset $


A point $p$ is an interior point of $E$ if there exists a basis element $B$ such
that $p \in B \subset E$.


$E^o =$ the set of all interior points 
\u
\hskip 15pt
$= \{x \in E ~|~$ there exist $B \in {\cal B}$ s. t.  $x \in B \subset E\}$
\u
\hskip 15pt= largest open set contained in
$E$ = $\cup_{U^{open} \subset E} U $.


Note: $E^o \subset E$.


$E$ is open iff every point of $E$ is an interior point.
\u
$E$ is open iff $E = E^o$.
\u
$E$ is open  iff $bd E \subset E^c$


$\overline{E} = E \cup E'$ = smallest closed set containing $E$ =
$\cap_{F^{closed}\supset E} F
$             
\u
\hskip 11pt$ 
= \{x \in X ~|~$  for all $U$ open such that $x \in U$, 
$U \cap
E \not= \emptyset \}$

$E$ is closed iff $E^c$ is open.
\u
$E$ is closed iff $E = \overline{E}$.
\u
$E$ is closed iff $E' \subset E$.
\u
$E$ is closed iff $bd E \subset E$


\vskip 10pt
\hrule

Subspace topology:  Suppose $Y \subset X$.

$E$ is open in $Y$ if and only if there exists a set $U$ open in $X$ such
that $E = U \cap Y$.

$E$ is closed in $Y$ if and only if there exists a set $F$ closed in $X$
such that $E = F \cap Y$.


Questions to consider:

Can a point be both a boundary point and an isolated point?

Can a point be both a boundary point and a limit point?

Can a point be both a boundary point and an interior point?

Can a point be both a limit point and an isolated point?

Can a point be both a limit point and an interior point?

Can a point be both an isolated point and an interior point?


Consider the integers, rationals, $\{{1 \over n} ~|~ n = 1, 2, 3, ... \}$, (0, 1), (0, 1], [0, 1], 
using standard, subspace, discrete, indiscrete topologies.


Give an exmple to show that a sequence can have a limit, but when the sequence is considered as a 
set, the set has no limit points.

Prove that $T_1$ is a needed part of the hypothesis of thm 17.9 and Hausdorff is a needed part of 
the hypothsesis of thm 17.10.


\end