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A. Miller \\
M551 \\
 Spring 2001\\
Exam 2\\
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Prove the following statements.

Please use a separate sheet of paper for each proof.

Please do not hand your exam in early.  Rewrite all your solutions
carefully inserting the reasons why everything you say is true.

\bigskip

\prob  If $f:X\to Y$ is continuous and onto and $X$ is connected,
then $Y$ is connected.

\prob  Suppose $A\subseteq X$, $A$ is compact (in the subspace topology),
and $X$ is a Hausdorff space.  Then $A$ is closed.

\prob  The closed unit interval $[0,1]$ is compact.

\prob Suppose that $X$ is a compact metric space and
$\langle x_n\in X\;:\;n\in\omega\rangle$ is a sequence.  Then
there exists a subsequence $\langle x_{k_n}\;:\;n\in\omega\rangle$
which converges to some $x\in X$.

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