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\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{\vskip -5pt}


Let $\sim$ be an equivalence relation on $X$.  Let $\pi: X \rightarrow 
X/\sim$, $\pi(x) = [x] = \{ y~|~ y \sim x \}$

$[A] = \cup_{a \in A} [a]$


Defn: $\sim$ is {\it open} if $U^{open} \subset X$ implies $[U]$ open in 
$X/\sim$.

Lemma: $\sim$ is { open} iff $\pi$ is an open mapping.

Lemma: If $\sim$ is {open} and if $X$ has a countable basis, then $X/\sim$  has a countable basis.

Lemma:  Let $\sim$ be {open}.  Then $\{ (x, y) ~|~ x \sim y \}$ is closed in $X \times X$ iff $X/\sim$  is Hausdorff.

Ex:  $P^n(\R) = RP^n$ =  n-dimensional real projective space is a smooth manifold.
 
 $RP^n = S^n /(x \sim -x) = (\R^n - \{0\})/ (\x \sim t\x)$ 

\end