\input amstex

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\def\ctln{\centerline}
\def\msk{\medskip}
\def\bsk{\bigskip}
\def\ssk{\smallskip}
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\def\ubr{\underbar}

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\ctln{\bf Math 871 Problem Set 3}

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Starred (*) problems are due Thursday, Sept. 15.

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\item{(*)} 16. [A useful openness test.] If $(X,{\Cal T})$ is a topological space, 
and $A\subseteq X$,  show that $A\in {\Cal T}$ if and only if

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\ctln{for all $x\in A$, there is a $U\in {\Cal T}$ so that $x\in U\subseteq A$}

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\item{17.} Show that if $f,g:X\ra Y$ are two functions from the topological
space $(X,{\Cal T})$ to $Y$, then the finest topology on $Y$ which makes
both functions continuous is the intersection of the finest topologies
making each function alone continuous.

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\item{18.} Show that if ${\Cal B}$ and ${\Cal B}^\prime$ are both bases for topologies on $X$,
then the set

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\centerline{${\Cal B}^{\prime\prime}$ = $\{B\cap B^\prime$ : $B\in{\Cal B}$ and $B^\prime\in{\Cal B}^\prime\}$}

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\item{} is also a basis for a topology on $X$ , and that the topology it generates is the 
coarsest topology on $X$ containing both ${\Cal B}$ and ${\Cal B}^\prime$ .

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\item{(*)} 19. [Munkres, p.83, Problem \#5] Show that if ${\Cal B}$ is a basis for a 
topology on $X$, then ${\Cal T}({\Cal B})$ is the intersection of all topologies 
that contain ${\Cal B}$. Show that the analogous statement is true for the topology
generated by a subbasis.

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\item{20.} Show that, for any set $X$, the set

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\centerline{$\Cal B$ = $\big\{B\subseteq X : X\setminus B$ is infinite $\bigr\}
\cup\bigl\{ X \bigr\}$}

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is a basis for a topology on $X$. What (familiar!) topolog(ies) does it generate?

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\item{21.} [Munkres, p.83, Problem \#8] (a) Show that the collection
${\Cal B} = \{(a,b)\ :\ a,b\in{\Bbb Q}\}$ is a basis, which generates
the `usual' topology on ${\Bbb R}$.

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\item{} (b) Show, by contrast, that the collection 
${\Cal B}^\prime = \{[a,b)\ :\ a,b\in{\Bbb Q}\}$ is a basis, 
but the topology it generates is strictly coarser 
than the `lower limit' topology ${\Cal T}_\ell$ on ${\Bbb R}$.

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\item{(*)} 22. Show that ${\Cal B} = \{(a,\infty)\times(b,\infty) : a,b\in{\Bbb R}\}$ is a basis for a topology ${\Cal T}$
on ${\Bbb R}^2={\Bbb R}\times {\Bbb R}$, which is coarser than the usual Euclidean topology on ${\Bbb R}^2$.
Show that ${\Cal B}^\prime = \{[a,\infty)\times[b,\infty) : a,b\in{\Bbb R}\}$ is a basis for a topology ${\Cal T}^\prime$ which is 
strictly finer than ${\Cal T}$, and not comparable to the usual Euclidean topology.

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\item{23.} [Munkres, p.92, Problem \#5] If ${\Cal T}\subseteq{\Cal T}^\prime$ are topologies on the
set $X$ and ${\Cal O}\subseteq{\Cal O}^\prime$ are topologies on the
set $Y$, show that the product topology ${\Cal T}\times{\Cal O}$ on $X\times Y$ is coarser
than the topology ${\Cal T}^\prime\times{\Cal O}^\prime$. Is the converse result true 
[i.e., product topology coarser implies that the topologies on each factor are coarser]?

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\item{24.} Show that if $(X,d)$ and $(Y,d^\prime)$ are metric spaces, then the product
topology on $X\times Y$ is also a metric topology. [There are lots of (correct) choices of metric on 
$X\times Y$; you can take your cue from ${\Bbb R}^2$ .]
\vfill
\end


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