\documentclass[minion]{homework651}
\include{hwextras}

\def\calB{\mathcal{B}}
\DeclareMathOperator{\Int}{\mathrm{Int}}

\doclabel{Math F651: Homework 3}
\docdate{Due: February 7, 2025}
%\docauthor{Your Name Here}

\begin{document}

Note: The book has Exercises, which are interspersed among the
prose, and Problems, which appear at the ends of the chapters.
It can be easy to confuse the two.  Exercises are denoted in blue.

\begin{problems}

\problem Suppose $\calB_1$ and $\calB_2$ are bases for topologies $\tau_1$
and $\tau_2$.  Show that $\tau_1\subseteq \tau_2$ if and only if
for every $B_1\in \calB_1$ and every $x\in B_1$ there is a $B_2\in \calB_2$
such that $x\in B_2\subseteq B_1$.

\problem Given a family $\{\tau_\alpha\}_{\alpha\in I}$ of topologies in $X$,
show that there is a unique smallest topology containing each $\tau_\alpha$.
Show also that there is a unique largest topology contained in each $\tau_\alpha$.
Take advantage of past work!

\problem Let $\calB=\{[a,b):a,b\in \Rats\}$.  Show that $\calB$ 
is a pre-basis
and hence generates a topology $\tau_\calB$.  Compare this topology to the 
lower-limit topology $\tau_\ell$.  In particular, determine if it is finer or coarser
or neither or both.

\problem  Problem 2-12

Conclude that the topology generated by a pre-basis $\mathcal B$
is the smallest topology in which every set from $\mathcal B$ is open.

\problem  Problem 2-15

\problem  Let $A$ be a subset of a topological space $X$, and let $\calB$ be
a basis for the topology.
\begin{subproblems}
\item Show that $x\in\overline{A}$ if and only if for every $B\in\calB$ with $x\in B$, $B\cap A\neq\emptyset$.
\item Show that $x\in\partial A$ if and only if for every $B\in\calB$  with $x\in B$, $B\cap A\neq\emptyset$
and $B\cap A^c\neq \emptyset$
\item Show that $\Int(A)\cap\partial A = \emptyset$ and $\overline{A}=\Int(A)\cup \partial A$.
\end{subproblems}
You are not allowed to cite Proposition 2.8 to prove these results.

\problem Problem 2-20

\end{problems}

\end{document}
