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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:
\v
(1) $f$ is continuous.
\v
(2) For every subset $A$ of $X$, $f(\overline{A}) \subset \overline{f(A)}$.
\v
(3) For every closed set $B$ of $Y$, $f^{-1}(B)$ is closed in X.
\v

(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. 

Defn:  A property of a space $X$ which is preserved by homeomorphisms
is called a topological property of $X$.

Defn:  $f: X \rightarrow Y$ is an imbedding of $X$ in $Y$ iff $f: X
\rightarrow f(X)$ is a homeomorphism.



Thm 18.2
\v
(a.) (Constant function)  The constant map \break
$f: X \rightarrow Y$,
$f(x) = y_0$ is continuous.
\v

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

\v

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

\v

(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.

\v

(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.

\v

(f.) (Local formulation of continuity)  If \break
$f: X \rightarrow Y$ 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.

\vskip 8pt

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 
$h:X
\rightarrow Y$,

\centerline{$h(x) = \cases{f(x) & $x \in A$ \cr g(x) & $x \in B$}$ is
continuous.}

\vskip 8pt

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

\eject

Defn:  A {\it group} is a set, G, together with a function $*: G \times G \rightarrow G$, $*(a, b) = 
a*b$ such that
\w
(0)  Closure:  $\forall a, b \in G$, $a*b \in G$.
\w

(1)  Associativity:  $\forall  a, b, c \in G$, 
	
\centerline{$(a * b) * c = a * (b * c)$.}
\w

(2) Identity: $\exists$ $e \in G$, such that $\forall a \in G$, 

\centerline{$e* a = a * e = a$.}
\w

(3) Inverses:  $\forall a \in G$, $\exists a^{-1} \in G$ such that 

\centerline{$a * a^{-1} = a^{-1}*  a = e$.} 

\vfill 

Defn:  A group $G$ is {\it commutative} or {\it abelian} if $\forall a, b \in G$, $a*b = b*a$.

\vfill
Defn:  A {\it topological group} is a set, G, such that
\w

(1)  $G$ is a group.
\w

(2)  $G$ is a topological space which is $T_1$.

\w
(3)  $*: G \times G \rightarrow G$, $*(a, b) =
a*b$ \hfil \break
and $i:G \rightarrow G$, $i(g) = g^{-1}$   are both continuous.



\end

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: