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\ctln{\bf Math 970 Homework 10}

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\ctln{Due Tuesday, Dec. 2}

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\item{43.} Show that a compact metric space $(X,d)$ is second countable.

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\item{} (Hint: look at $\{ N_d(x,1/n) : x\in X\}$ for each $n$ .)

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\item{44.} Show that a closed subset of a Lindel\"of space is Lindel\"of.

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\item{45.} Show by example that a closed subset of a separable space 
need not be separable.

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\item{46.} Show  that the continuous image of a separable space 
is separable, and the continuous image of a Lindel\"of space is Lindel\"of..

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\item{47.} Show that if $(X,{\Cal T}({\Cal C}))$ is second countable 
(with ${\Cal C} = \{ C_n\}_{n=1}^\infty$ 
countable), then \underbar{every}
basis ${\Cal B}$ for ${\Cal T} = {\Cal T}({\Cal C})$
contains a countable basis ${\Cal B}^\prime\subseteq{\Cal B}$.

\item{} (Hint: look at all $B\in{\Cal B}$ with $C_m\subseteq B\subseteq C_n$ for some $m,n$ ;  then pick (at most) one for each pair...)

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