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17.  Closed Sets and Limit Points

Defn:  The set $A$ is {\bf closed} iff $X - A$ is open.

Thm 17.1:  $X$ be a topological space if and only if the following conditions hold:
\v
(1) $\emptyset$, $X$ are closed.
\v
(2) Arbitrary intersections of closed sets are closed.
\v
(3) Finite unions of closed sets are closed.

Note arbitrary intersections of open sets need not be open.  Example:
$\cap_{n=1}^\infty (-{1 \over n}, {1 \over n})$ = 

Note arbitrary unions of closed sets need not be closed.  Example:
$\cup_{n=1}^\infty [{1 \over n}, 1 - {1 \over n}]$ = 


Thm 17.2:  Let $Y$ be a subspace of $X$.  Then a set $A$ is closed in $Y$ if and only if it equals the intersection of a closed set of $X$ with $Y$.

Thm 17.3:  Let $Y$ be a subspace of $X$.  If $A$ is closed in $Y$ and $Y$ is closed in $X$, then $A$ is closed in $X$.

Def:  The {\bf interior} of $A$ = $Int~A$ = $A^0$ = $\cup_{U^{open} \subset
A}U$

Def:  The {\bf closure} of $A$ = $Cl~A = \overline{A} = \cap_{A \subset
F^{closed}} F$

Note:  $\overline{A}$ is the smallest closed set containing $A$.

Thm 17.4:  Let $Y$ be a subspace of $X$, $A \subset Y$.  Let $\overline{A}$
denote the closure of $A$ in $X$.  Then the closure of $A$ in $Y$ equals
$\overline{A} \cap Y$.

Defn:  $A$ {\bf intersects} $B$ if $A \cap B \not= \emptyset$

Thm 17.5:  Let $A$ be a subset of the topological space $X$.

(a) $x \in \overline{A}$ if and only if ($x \in U^{open}$ implies 

\rightline{$U
\cap A \not=\emptyset$).}

(b) $x \in \overline{A}$ if and only if ($x \in B$ where $B$ is a basis 

\rightline{element implies $B \cap A \not=\emptyset$).}

Defn:  $U$ is a {\bf neighborhood} of $x$ if $U$ is an open set containing $x$.
\eject

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 $X$ be $T_1$, $A \subset X$.  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
 \end
Ex
 155-741-110-72 
155-734-740-72 

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.

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 of points of $X$ converges to at most one point of $X$.  (IFF?)

${1 \over n}$ converges to in the finite complement topology.

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: