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18. Continuous Functions



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  

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

\v

(g)  (The pasting lemma) If \hfil \break
$f: X \rightarrow Y$ and
$X = \cup_{i=1}^n A_i $, $A_i$ closed where $f|_{A_i}: A_i
\rightarrow Y$ 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.

Note the above also holds for arbitrary products in the product topology.

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Thm 36.1 (Existence of partitions of unity).  Let $\{U_1, ..., U_n\}$ be a 
finite indexed open cover of $X$ and let $X$ be $T_4$.  Then there exists 
a  partition of unity dominated by $\{U_i\}$

Thm 36.2  If $X$ is a compact $m$-manifold, then $X$ can be imbedded in 
$R^N$ for some positive integer $N$.



\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.


\end