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\begin{flushright}
 A. Miller \\
 M551 \\
 Spring 2001\\
\end{flushright}


\prob Define
$\Pi_{i\in I}X_i=
\{\langle x_i:i\in I\rangle\;:\;\forall i\in I\; x_i\in X_i\}$

\bigskip
The $I$-sequence $\langle x_i:i\in I\rangle$ which is also
written $(x_i)_{i\in I}$, is a function
with domain $I$ which maps each $i\in I$ to an element $x_i\in X_i$.

\prob If $X_i$ for $i\in I$ are topological spaces, then
what is a basic open subset of $\Pi_{i\in I}X_i$ (in the Tychonoff
product topology)?

A set of the form $\Pi_{i\in I}U_i$ where $U_i\subseteq X_i$ is open for
every $i\in I$ and $U_i=X_i$ for all but finitely many $i\in I$.

\prob If $X_i$ for $i\in I$ are topological spaces, then
what is a basic open subset of $\Box_{i\in I}X_i$ (in the Box
product topology)?

A set of the form $\Pi_{i\in I}U_i$ where $U_i\subseteq X_i$ is open
for every $i\in I$.

\prob A metric $d$ on $X$ is a function $d:X\times X\to [0,\infty)$ such
that
\par (a) for all $x,y\in X\;\;\;d(x,y)=0$ iff $x=y$,
\par (b) for all $x,y\in X\;\;\;d(x,y)=d(y,x)$, and
\par (c) for all $x,y,z\in X\;\;\;d(x,z)\leq d(x,y)+d(y,z)$

\prob If $d$ is a metric on $X$ and $\epsilon>0$, then
$B^d_\epsilon(x)=\{y\in X\;:\; d(x,y)<\epsilon\}$

\prob If $(X,d)$ is a metric space then $U\subseteq X$ is
open
\par iff  $\forall x\in U\;\;\;\exists \epsilon>0\;\;
B^d_\epsilon(x)\subseteq U$

\prob For a sequence $\langle x_n:n\in\omega\rangle$ in a space $X$
then $\lim_{n\to\infty}x_n=x_0$ iff for all $U$ an open neighborhood
of $x_0$ for all but finitely many $x_n\in U$.

\prob For $X$ a topological space, $A\subseteq X$ is clopen iff
$A$ is closed and open.

\prob For $X$ a topological space, $X$ is connected iff it
contains no nontrivial clopen set.  ($X$ and the empty set are
the trivial clopen sets.)

\prob ${\cal U}$ is an open cover of $X$ iff the elements of
${\cal U}$ are open and $\bigcup{\cal U}=X$.

\prob ${\cal V}$ is a subcover of the cover ${\cal U}$ iff
${\cal V}\subseteq {\cal U}$ and $\bigcup{\cal V}=X$.

\prob A topological space $X$ is compact iff every open cover
of $X$ has a finite subcover.



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