# MAT 364: Topology and Geometry C

Welcome to MAT 364 --- Topology and Geometry.

This is an introductory course to "point set topology" and to "algebraic
topology". We will study topologies arrising the metric spaces, and more
general topologies. One the main problems in topology is to determine
whether or not two topological spaces are "topologically equivalent".
Towards this end with will study "compactness", "connectedness", the
Hausdorff property and other elmentary properties of topological spaces.
We shall also study the "fundamental group" of a topological space and
develop some tools for computing this important group.
This is a "proof course", so each student should try to understand all
the proofs that we discuss and develop (thru hard work and practice) a
skill in proving elementary claims.

### Text Book

###
{\it Topology of Surfaces\} by L. Christine Kinsey, Spring (1993).

### Exams

The final exam for Mat 364 will be held on December 20, 11:00am-1:30pm.
There will also be one inclass midterm on November 1.

### Homework

###

Homework will be assigned every week on this web site. Please turn it in
during the first class of the following week.

### Grading

###

Homework=50%

Midterm=20%

Final Exam=30%

### Course instructor

**Lowell Jones** (course coordinator), lejones@math.sunys.edu,
Math Tower 2-111. Telephone: 632-8248.

Office hours: Monday 12:00-1:00pm
in Math undergraduate office (P-143); Tuesday and Thursday 11:30-12:30 in
2-111.
### Grader

**Ya Sle Cha**, ycha@math.sunysb.edu,
Math Learning Center (offices in the back). Office hours:?.
### Disabilities

If you have a physical, psychological, medical or
learning disability that may impact on your ability
to carry out assigned course work, please
contact the staff in the Disabled Student Services office
(DSS), Room 133 Humanities, 632-6748/TDD. DSS will review
your concerns and determine, with you, what accommodations
are necessary and appropriate. All information and
documentation of disability is confidential.
### Homework II (due on Tuesday 9/25)

### Disabilities

If you have a physical, psychological, medical or
learning disability that may impact on your ability
to carry out assigned course work, please
contact the staff in the Disabled Student Services office
(DSS), Room 133 Humanities, 632-6748/TDD. DSS will review
your concerns and determine, with you, what accommodations
are necessary and appropriate. All information and
documentation of disability is confidential.
Do \#2.27,2.28 on page 28,29 of text.
\vspace{.1in}
Also complete the following two problems.
\vspace{.1in}
Let $X$ denote a metric space with metric
$d:X\times X\longrightarrow X$.
By an "open covering" of X we mean a collection $\{U_{i}:i\in I\}$ of open
subsets $U_{i}\subset X$ such that $X=\bigcup_{i\in I}U_{i}$.
By a "countable open cover" of $X$ we mean an open covering for $X$
where the index set $I$ is equal to the natural numbers $\{1,2,3,...\}$.
A "finite open cover" for $X$ is an open covering $\{U_{i}\mid i\in I\}$
for $X$ where the index set $I$ is finite. If $\{U_{i}\mid i\in I\}$ and
$\{V_{j}\mid j\in J}$ are two open covers for $X$ then we say that the second
of these open covers is a "subcover" of the first if for each
$j\in J$ there is $i\in I$ such that $V_{j}=U_{i}$; this subcover is called
a "countable (or finite) subcover" if it is a countable (or finite) open
covering of $X$.
\vspace{.1in}
{\bf (1)} Show that every countable open cover for $X$ contains a finite
subcover.
\vspace{.1in}
{\bf (2)} Let $f:X\longrightarrow X^{\prime}$ denote a map between metric
spaces $X,X^{\prime}$ equipped with metrics $d,d^{\prime}$ respectively.
Show that $f$ is continuous iff for any convergent sequence
$\{x_{i}\mid i=1,2,3,...\}$ in $X$ the image sequence
$\{f(x_{i})\mid i=1,2,3,...\}$ is convergent in $X^{\prime}$
and
$$f(limit_{i\rightarrow \infty}x_{i})=limit_{i\rightarrow \infty}f(x_{i})\hspace{1em}.$$