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

Define the following concepts: (Answers have been added after ``:'')


\prob Countable: A set $X$ is countable iff it is empty or
there is a map from $\omega$ onto $X$.

\prob Uncountable: Uncountable means not countable.

\prob $\omega_1\;\;\;$ (or as Munkres calls it $S_{\Omega}$):
\par\noindent This is the smallest uncountable well-ordering.  It is
characterized by the properties
\par (a) it is a well-ordering
\par (b) it is uncountable
\par (c) each initial segment is countable, i.e. $\{\alpha:\alpha<\beta\}$
is countable for each $\beta<\omega_1$.


\prob Dense: A set $D\subseteq X$ is dense iff its closure is $X$.

\prob Separable: A space is separable iff it contains a countable
dense subset.

\prob Neighborhood basis: A neighborhood basis for $x\in X$ is
a family ${\cal B}$ of open neighborhoods of $x$ such that
for any open neighborhood $U$ of $x$ there exists
$B \in {\cal B}$ with $x\in B\subseteq U$.

\prob First countable: A space is first countable iff each of
its points has a countable neighborhood base.

\prob Second countable: A space is second countable iff it
has a countable basis.

\prob Lindel\"{o}f: A space is Lindel\"{o}f iff every open cover contains
a countable subcover.

\prob Hausdorff: A space is Hausdorff iff any two distinct points
can be separated by disjoint open sets.

\prob Regular: A space $X$ is regular iff it is Hausdorff and for every
closed set $C$ and point $x\in X-C$ there
exists disjoint open $U,V$ with $x\in U$ and $C\subseteq V$. Equivalently,
for every $x\in U$ open there exists open $V$ with $x\in V\subseteq
\overline{V}\subseteq U$

\prob Normal: A space is normal iff it is Hausdorff and any two
disjoint closed sets can be separated by disjoint open sets.

\prob n-Manifold: An n-manifold is a  second countable
Hausdorff space
which has the property that every point has a neighborhood
homeomorphic to ${\mathbb R}^n$.

\prob Quotient map and Quotient space:
$Y$ is a quotient space of
$X$ iff there exists an onto map $p:X\to Y$ which
satisfies for all $U\subseteq Y$:
\par $U$ is open in $Y$ iff $p^{-1}(U)$ is open in $X$.
\par\noindent Such a  $p$ is called a quotient map.

\prob (Extra Credit) Embedding or imbedding or emersion or immersion
or submersion:

$X$ can be embedded into $Y$ iff $X$ is a homeomorphic to a subspace of $Y$.


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