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Defn:  $x \in X$ is a {\bf limit point} of $A$ iff $x \in U^{open}$ implies $U
\cap A - \{x\} \not=\emptyset$.

Defn:  $A'$ = the set of all limit points of $A$.

Thm 17.6:  $\overline{A} = A \cup A'$.

Cor 17.7:  $A$ closed if and only if $A' \subset A$.

Defn:  $x_n$ converges to a limit $x$ if for every neighborhood $U$ of $x$, there exists a positive integer $N$ such that $n \geq N$ implies $x_n \in U$.

Note:  limit point of a set is not the same as limit of a sequence.

Defn:  $X$ is {\bf Hausdorff space} 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.

Defn:  $X$ is $T_1$ if every one point set is closed.

Defn:  $X$ is $T_1$ if $\forall$ $x_1, x_2 \in X$ such that $x_1 \not= x_2$, 
$\exists$ 
nbhds $U_1$ and $U_2$ of $x_1$ and $x_2$, respectively, such that $x_2 \not\in U_1$ and $x_1 
\not\in U_2$.

Defn:  $X$ is $T_1$ if $\forall$ $x_1 \in X$,  $x_2 \not= x_1$ implies
$\exists$ a
nbhd $U$ of $x_1$ such that $ x_2 \not\in U$.

Defn:  $X$ is $T_0$ if $\forall$ $x_1, x_2 \in X$ such that $x_1 \not= x_2$, 
$\exists$ EITHER 
[a nbhd $U$ of $x_1$ such that $x_2 \not\in U$]
or [a nbhd $V$ of $x_2$ such that $x_1 \not\in V$]


%%Defn:  $X$ is $T_0$ if $\forall$ $x_1, x_2 \in X$ such that $x_1 \not= x_2$, $\exists$ a nbhd $U$ of 
%%$x_i$ for $i = 1$ OR for $i = 2$ such that $x_j \not\in U$ for {\xout{$j \not= i$,}} $j = 2$ or $j = 
%%1$.


Thm 17.9:  Let $X$ by $T_1$, $A \subset X$.  Then $x$ is a limit point of $A$ if
and only if every neighborhood of $x$ contains infinitely many points of $A$.

Thm 17.10:  If $X$ is Hausdorff, then a sequence 

\line{of points of $X$ converges to at most one point of 
$X$.} 


Thm 17.11:  If $X$ has the order topology, then $X$ is Hausdorff.  The product of two Hausdorff spaces is Hausdorff.  A subspace of a Hausdorff space is Hausdorff.

\end

18. Continuous Functions

Defn:  $f^{-1}(V) = \{x ~|~ f(x) \in V \}$.

Defn:  $f: X \rightarrow Y$ is continuous iff for every $V$ open in $Y$, $f^{-1}(V)$ is open in $X$. 

Lemma: $f$ continuous if and only if for every basis element $B$, $f^{-1}(B)$ is open in $X$. 

Lemma: $f$ continuous if and only if for every subbasis element $S$, $f^{-1}(S)$ is open in $X$. 

Thm 18.1:  Let $f: X \rightarrow Y$.  Then the following are equivalent:

(1) $f$ is continuous.

(2) For every subset $A$ of $X$, $f(\overline{A}) \subset \overline{f(A)}$.

(3) For every closed set $B$ of $Y$, $f^{-1}(B) is closed in X.


(4) For each $x \in X$ and each neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U) \subset V$.

Defn:  $f: X \rightarrow Y$ is a homeomorphism iff $f$ is a bijection and both $f$ and $f^{-1}$ is continuous. 

topological property

imbedding

Thm 18.2

(a.) (Constant function)  The constant map $f: X \rightarrow Y$, $f(x) = y_0$ is continuous.

(b.) (Inclusion)  If $A$ is a subspace of $X$, then the inclusion map $f: A \rightarrow X$, $f(a) = a$ is continuous.

(c.) (Composition) If $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are continuous, then $g \circ f: X \rightarrow Z$ is continuous.

(d.) (Restricting the Domain) If $f: X \rightarrow Y$ is continuous and
if $A$ is a subspace of $X$, then the restricted function $f|_A: A
\rightarrow Y$, $~f|_A(a) = f(a)$ is continuous.

(e.) (Restricting or Expanding the Codomain)  If $f: X \rightarrow Y$ is continuous and if $Z$ is a subspace of $Y$ containing the image set $f(X)$ or if $Y$ is a subspace of $Z$, then $g: X \rightarrow Z$ is continuous.

(f.) (Local formulation of continuity)  If $f: X \rightarrow Y$ and $X = \cup U_\alpha$, $U_\alpha$ open where $f|_{U_\alpha} U_\alpha \rightarrow Y$ is continuous, then $f: X \rightarrow Y$ is continuous.

Thm 18.3 (The pasting lemma)> Let $X = A \cup B$ where $A$, $B$ are
closed in $X$.  Let $f: A \rightarrow Y$ and $g: B \rightarrow Y$ be
continuous.  If $f(x) = g(x)$ for all $x \in A \cap B$, then $h:X
\rightarrow Y$, $h(x) = \cases{f(x) & $x \in A$ \cr g(x) & $x \in B$}
is
continuous.

THm 18.4:  Let $f: A \rightarrow X \times Y$ be given by the equations
$f(a) = (f_1(a), f_2(a))$ where $f_1: A \rightarrow X$, $f_2: A
\rightarrow X$.  Then $f$ is continuous if and only if $f_1$ and $f_2$ are continuous.

19. The Product Topology.

Defn:  Let $J$ be an index set.  Given a set $X$, a {\bf J-tuple} of elements of $X$ is a function ${\bf x}: J \rightarrow X$.  The {\bf $\alpha$th coordinate of x} = $x_\alpha$ = {\bf x}$(\alpha)$



\end

Let $\pi_j: \Pi_{i=0}^n X_i \rightarrow X_j$, $\pi_j(x_1, ..., x_n) = x_j$.
$\pi_j$ is the projection of $\Pi_{i=0}^n X_i $ onto the $j$th component.


Note: