% LaTex2e
\documentclass[12pt]{article}
\usepackage{amssymb,latexsym}
\pagestyle{empty}

% save paper style
\oddsidemargin -45pt \evensidemargin 5pt \marginparwidth 68pt
\marginparsep 10pt \topmargin -40pt \headheight 10pt
\headsep 15pt \footskip 35pt
\textheight = 645pt  \textwidth 440pt \columnsep 10pt \columnseprule 5pt

\newcount\probno
\probno=0
\def\prob{\bigskip\bigskip\bigskip\par\hrule\par\bigskip\bigskip
\bigskip\noindent
\advance\probno by 1 \par\noindent\the\probno .  }


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

Complete the following definitions:

\prob If ${\cal F}$ is a family of sets, then $\bigcup {\cal F}=$

\prob $\tau$ is a topology on $X$ iff

\prob The discrete topology on $X$ is

\prob The cofinite topology on $X$ is


\prob ${\cal B}$ is a basis for a topology on $X$ iff

\prob If ${\cal B}$ is a basis for a topology on $X$ then a
set $U\subseteq X$ is open iff

\prob ${\cal S}$ is a subbase for a topology on $X$ iff

\prob $(L,\leq)$ is a linear order iff

\prob The order topology on the linear order $(L,\leq)$ is

\prob $X\subseteq L$ and $b\in L$ is an upper bound of $X$ iff

\prob The linear order $(L,\leq)$ has the least upper bound property iff

\prob $(L,\leq)$ is a well-ordering iff

\prob The product topology on $X\times Y$ is

\prob The subspace topology on $Y\subseteq X$ is

\prob A set $C\subseteq X$ is closed iff

\prob The closure of the set $A\subseteq X$ is

\newpage

\begin{center}
  Answers
\end{center}
\probno=0
\def\prob{\advance\probno by 1 \par\noindent\the\probno . }

\prob If ${\cal F}$ is a family of sets, then $\bigcup {\cal F}=
\{x:\exists F\in {\cal F}\;\; x\in F\}$.

\prob $\tau$ is a topology on $X$ iff the empty set and $X$
are in $\tau$, $\tau$ is closed under arbitrary unions,
and $\tau$ is closed under finite intersections.  Equivalently,

$\emptyset,X\in \tau$

$\forall {\cal U}\subseteq \tau\;\; \bigcup {\cal U}\in \tau$

$\forall U,V\in \tau \;\; U\cap V\in \tau$

\prob The discrete topology on $X$ is the topology on $X$ in which all
subsets of $X$ are open.

\prob The cofinite topology on $X$ is the topology $\tau$ on $X$
\par $\tau=\{U\subseteq X: U=\emptyset $ or $X\setminus U$ is finite$\}$.

\prob ${\cal B}$ is a basis for a topology on $X$ iff
$\bigcup {\cal B}=X$ and
\par\noindent $\forall B_1,B_2\in {\cal B}\;\;\forall p\in B_1\cap B_2\;\;
\exists B_3\in {\cal B}\;\; p\in B_3\subseteq B_1\cap B_2$.

\prob If ${\cal B}$ is a basis for a topology on $X$ then a
set $U\subseteq X$ is open iff
\par $\forall p\in U\;\;\exists B\in {\cal B}
\;\; p\in B\subseteq U$.

\prob ${\cal S}$ is a subbase for a topology on $X$ iff $\bigcup {\cal S}=X$.

\prob $(L,\leq)$ is a linear order iff the following four axioms hold:
\par $\forall x\in L\;\; x\leq x$
\par $\forall x,y\in L\;\; (x\leq y$ and $y\leq x$) implies $x=y$
\par $\forall x,y,z\in L\;\; (x\leq y$ and $y\leq z$) implies $x\leq z$
\par $\forall x,y\in L\;\; x\leq y$ or $y\leq x$

\prob The order topology on the linear order $(L,\leq)$ is the topology
with the basis consisting of open intervals in $L$ (including open rays
$(-\infty,a)$ and  $(a,\infty)$).

\prob $X\subseteq L$ and $b\in L$ is an upper bound of $X$ iff
$\forall x\in X\;\;\; x\leq b$.

\prob The linear order $(L,\leq)$ has the least upper bound property iff
every nonempty $X\subseteq L$ which has an upper bound has an
upperbound which is $\leq$ all other upper bounds.

\prob $(L,\leq)$ is a well-ordering iff it is a linear order in
which all nonempty subsets have a least element, i.e., $\forall
X\subseteq L$ which is nonempty $\exists x\in X$ such that
$\forall y\in X\;\; x\leq y$.

\prob The product topology on $X\times Y$ is the topology
with basis
$${\cal B}=\{U\times V: U^{\rm open}\subseteq X,\;\; V^{\rm open}\subseteq Y\}$$

\prob The subspace topology on $Y\subseteq X$ is
$$\tau_Y=\{U\cap Y\;:\; U\in \tau_X\}$$

\prob A set $C\subseteq X$ is closed iff $X\setminus C$ is open.

\prob The closure of the set $A\subseteq X$ is
$$\bigcap\{C^{\rm closed}\subseteq X\;:\; A\subseteq C\}$$


\end{document}