# 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.

### 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.

### 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.

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 $$f(limit_{i\rightarrow \infty}x_{i})=limit_{i\rightarrow \infty}f(x_{i})\hspace{1em}.$$