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22. The Quotient Topology

Defn:  Let $X$ and $Y$ be topological spaces; let $p: X \rightarrow
Y$ be a surjective map.  The map $p$ is a {\bf quotient map} if $U$
is open in $Y$ if and only if $p^{-1}(U)$ is open in $X$.

Defn:  $C \subset X$ is {\bf saturated} with respect to $p$ if
\break $p^{-1}(\{y\}) \cap C \not= \emptyset$ implies 
$p^{-1}(\{y\}) \subset C$.  

That is, $C$ is saturated if there exists a set $D \subset Y$ such that $C = p^{-1}(D)$.  

That is, $C$ is saturated if $C = p^{-1}(p(C))$ 

Lemma: $p: X \rightarrow Y$ is a quotient map if and only if $p$ is continuous and $p$ maps saturated open sets of $X$ to open sets of $Y$.

Defn:  $f: X \rightarrow Y$ is an {\bf open map} if for every open set $U$ of $X$, $f(U)$ is open in $Y$.

Defn:  $f: X \rightarrow Y$ is a {\bf closed map} if for every closed set $A$ of $X$, $f(A)$ is closed in $Y$.

Lemma:  An open map is a quotient map.  A closed map is a quotient map.  There exist quotient maps which are neither open nor closed.

Defn:  Let $X$  be a topological spaces and let $A$ be a set; let $p: X \rightarrow Y$ be a surjective map.  The {\bf quotient topology} on $A$ is the unique topology on $A$ which makes $p$ a quotient map.

Defn:  A partition, $X^*$, of a set $X$ is a collection of disjoint
subsets of $X$ whose union is $X$.  I.e., \break
$X^* = \{C_\alpha ~|~
\alpha \in A \}$, $X^* \subset P(X)$, the power set on $X$, $C_\alpha
\cap C_\alpha' = \emptyset$, for all $\alpha, \alpha' \in A$, and $X
= \cup_{\alpha \in A} C_\alpha$.

Define $p: X \rightarrow X^*$, $p(x) = C_\alpha$ if $x \in C_\alpha$.
The quotient topology induced by $p$ on $X^*$ is the {\bf quotient
space} of $X$.  $X^*$ is called the {\bf identification space} or
{decomposition space} of $X$.

Thm 22.2:  Let $p:X \rightarrow Y$ be a quotient map.  Let 
$g:X \rightarrow Z$ be a map with is constant on each set $p^{-1}(y)$
for
all $y \in Y$.  Then $g$ induces a map $f:Y \rightarrow Z$ such that
$f \circ p = g$.  $f$ is continuous if and only if $g$ is continuous.
$f$ is a quotient map if and only if $g$ is a quotient map.

Cor 22.3:  Let $g:X \rightarrow Z$ be a surjective continuous map.
Let $X^* = \{g^{-1}(\{z\} ~|~ z \in Z\}$ with the quotient topology.

\v
(a.)  The map $g$ induces a bijective continuous map \break
$f: X^*
\rightarrow Z$, which is a homeomorphism if and only if $g$ is a
quotient map.
\v
(b.)  If $Z$ is Hausdorff, so in $X^*$.

\end

8.2:  Series of Positive and Negative Terms; Power series.

Defn:  The series $\Sigma_{n=1}^\infty u_n$ {\bf converges absolutely} if 
$\Sigma_{n=1}^\infty |u_n|$.  $\Sigma_{n=1}^\infty u_n$ {\bf converges conditionally} if $\Sigma_{n=1}^\infty u_n$ converges, but $\Sigma_{n=1}^\infty |u_n|$ diverges. 

Thm 8.6:  If $\Sigma_{n=1}^\infty |u_n|$ converges, then $\Sigma_{n=1}^\infty u_n$ converges.

Thm 8.7 (Alternating series theorem)  Suppose that $u_n$ satisifies the following:

(i) The $u_n$ are alternately positive and negative.

(ii) $|u_{n+1}| < |u_n|$

(iii) $lim_{n \rightarrow \infty} u_n = 0$

Then $\Sigma_{n=1}^\infty u_n$ converges. Furthermore,  $\Sigma_{n=1}^\infty u_n$ lies between $s_n$ and $s_{n+1}$ for all $n$.

Thm 8.9 (Root Test).  Suppose $$\overline{lim}_{n \rightarrow
\infty} (|u_n|)^{1 \over n} = r \hbox{ or } \overline {lim}_{n
\rightarrow \infty} (|u_n|)^{1 \over n} = +\infty$$ Then

(1) if $r > 1$, then $\Sigma_{n=1}^\infty u_n$ converges absolutely.

(2) if $r < 1$ or ${lim}_{n \rightarrow \infty}  (|u_n|)^{1 \over n} = +\infty $, then $\Sigma_{n=1}^\infty u_n$ diverges.

(3) If $r = 1$, the test gives no information.

Thm 8.8 (Ratio Test)  Suppose that $u_n \not= 0$ for all $n$, and
that $$\overline{lim}_{n \rightarrow \infty} |{u_{n+1} \over u_n}| =
r \hbox{ or } {lim}_{n \rightarrow \infty} |{u_{n+1} \over u_n}| =
+\infty$$ Then

(1)  if $r > 1$, then $\Sigma_{n=1}^\infty u_n$ converges absolutely.

(2) if $r < 1$ or ${lim}_{n \rightarrow \infty} |{u_{n+1} \over u_n}| = +\infty $, then $\Sigma_{n=1}^\infty u_n$ diverges.

(3) If $r = 1$, the test gives no information.

\hrule

Defn:  A {\bf power series} is a series of the form $\Sigma_{n=1}^\infty c_n(x - 1)^{n-1}

Lemma 8.1:  If the series $\Sigma_{n=1}^\infty u_n$ converges, then there
exists a number $N$ such that $|u_n| \leq M$ for all $n$.

I

\end