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Homework 1 (first part) [adapted from latex HW of Colin McKinney]

\noindent \begin{bf}1.0\end{bf}  Show 
$X - \bigcap_{i=1}^{n}U_i = \bigcup_{i=1}^{n}(X - U_{i})$


$x \in X - \bigcap_{i=1}^{n}U_i$ 

iff $x \in X$ and $x \not\in \bigcap_{i=1}^{n}U_i$ 

iff $x \in X$ and $x \not\in U_i$ for some $i$  

iff $x \in X -  U_i$ for some $i$  

iff $x \in \bigcup_{i=1}^{n}(X - U_{i})$.  


\noindent \begin{bf}1.1\end{bf} Let $X$ be a toplogical space; let $A$ be
a subset of $X$.  Suppose that for each $x \in A$ there is an open set 
$U_x$
containing $x$ such that $U_x \subset A$.  Show that $A$ is open in $X$.  

Proof:  For each  $x \in A$, take an open set $U_x$
containing $x$ such that $U_x \subset A$ ($U_x$ exists by hypothesis). 
Hence $\cup_{x \in A} U_{x}$ is also open (by defn of topology).  
$\cup_{x \in A} U_{x} = A$,
and so $A$ must also be open. 


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