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UF MTG 4302/5316\\
Introduction to Topology 1\\
Fall 2023
\begin{center}
{\large \bf Homework 2}\\
Due Monday, September 18, anytime, on Canvas
\end{center}

{\bf Reading.}
Munkres \S10 -- \S15


{\bf Problems.}
\begin{itemize}

\item \S9 \#5. 

(a) Use the choice axiom to show that if $f\colon A\to B$ is surjective, then $f$ has a right inverse $h\colon B\to A$.

(b) Show that if $f\colon A\to B$ is injective and $A$ is not empty, then $f$ has a left inverse.
Is the axiom of choice needed?

\emph{If $f\colon A \to B$ is a function, then a \emph{right inverse} is a function $h \colon B \to A$ satisfying $f(h(b))=b$ for all $b \in B$, and a \emph{left inverse} is a function $g\colon B \to A$ satisfying $g(f(a))=a$ for all $a \in A$.}

\item \S11 \#8.
A typical use of Zorn's lemma in algebra is the proof that every vector space has a basis.
Recall that if $A$ is a subset of the vector space $V$, we say a vector belongs to the \emph{span} of $A$ if it equals a finite linear combination of elements of $A$.
The set $A$ is \emph{independent} if the only finite linear combination of elements of $A$ that equals the zero vector is the trivial one having all coefficients zero.
If $A$ is independent and if every vector in $V$ belongs to the span of $A$, then $A$ is a \emph{basis} for $V$.

(a) If $A$ is independent and $v\in V$ does not belong to the span of $A$, show $A\cup \{v\}$ is independent.

(b) Show the collection of all independent sets in $V$ has a maximal element.

(c) Show that $V$ has a basis.

\item \S13  \#4(a).
If $\{\tau_\alpha\}$ is a family of topologies on $X$, show that $\cap \tau_\alpha$ is a topology on $X$.
Is $\cup \tau_\alpha$ a topology on $X$?

\end{itemize}

\vfill

{\bf Recommend Problems (not to turn in).}
\begin{itemize}

\item \S9 \#6.

\item \S11 \#2; this question shows that partially ordered sets defined using the axioms for $\prec$ are really the same mathematical object as partially ordered sets defined using the axioms for $\preceq$.

\item \S13 \#1.

\end{itemize}

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