## COURSE DESCRIPTION

Math 109a is the first course in the Ma 109 sequence, Introduction to Geometry and Topology.  In the first part of the course, we will introduce notions of general point-set topology, basic examples and constructions.  Topics will include the notions of compactness, metrizability, separation properties, and completeness. The second part of the course will be an introduction to algebraic topology, including the fundamental group and homology theory.

## PREREQUISITES

Specifically, you should be comfortable with the following topics:

1. Real analysis: limits, continuity, differentiation, and integration.
2. Linear algebra: vector spaces, matrices, linear transformations, determinants, and eigenvalues/eigenvectors.
3. Abstract algebra: groups, rings, and fields.
4. Set theory: basic notions of sets, relations, and functions.

In summary, a strong background in calculus, linear algebra, abstract algebra, and set theory, along with good problem-solving skills and abstract thinking abilities, are the typical prerequisites for a topology course.

## INSTRUCTORS

Hyun Chul Jang

Harry Bateman

Postdoctoral Scholar Teaching Fellow in Mathematics

Problem 1.

We say a space $X$ is contractable if the identity map id : $X \rightarrow X$ is null-homotpoic.
(1) Show that $\mathbb{R}^n$ is contractable for any $n \geqslant 0$.
(2) Show that a contractable space is path connected.
(3) Let $\left{X_\alpha\right}_{\alpha \in \mathcal{J}}$ be a family of contractable spaces. Is $X:=\Pi_{\alpha \in \mathcal{J}} X_\alpha$ contractable? Prove or provide a counterexample.

Proof .

(1) To show that $\mathbb{R}^n$ is contractable, we need to find a homotopy $H : \mathbb{R}^n \times [0,1] \rightarrow \mathbb{R}^n$ such that $H(x,0) = x$ and $H(x,1) = c$ for some constant $c \in \mathbb{R}^n$. One possible homotopy is $H(x,t) = tc + (1-t)x$ for any $c \in \mathbb{R}^n$. It is easy to check that $H$ is continuous, $H(x,0) = x$, and $H(x,1) = c$ for any $x \in \mathbb{R}^n$.

(2) Suppose $X$ is contractable and let $f, g : [0,1] \rightarrow X$ be two paths between $x$ and $y$ in $X$. Then the concatenation of $f$ and the reverse of $g$ gives a loop $\gamma$ based at $x$. Since $X$ is contractable, there exists a homotopy $H : X \times [0,1] \rightarrow X$ between the identity map on $X$ and a constant map $c$. In particular, $H(x,1) = c$ for any $x \in X$. Therefore, the map $F : [0,1] \times [0,1] \rightarrow X$ defined by $F(s,t) = H(f(s),t)$ is a homotopy between $f$ and $g$. Hence, $X$ is path connected.

(3) The product space $X:=\Pi_{\alpha \in \mathcal{J}} X_\alpha$ need not be contractable even if each $X_\alpha$ is contractable. For example, let $\mathcal{J} = {0,1}$ and $X_0 = X_1 = {0,1}$ with the discrete topology. Then $X_0$ and $X_1$ are contractable, but $X$ is not since the map $id_X : X \rightarrow X$ is not null-homotopic. To see this, note that any homotopy $H : X \times [0,1] \rightarrow X$ between $id_X$ and a constant map must fix the first coordinate since $X_0$ is contractable. Therefore, $H(x,1) = x$ for any $x \in X$, but $H(0,0) \neq H(1,0)$, so $H$ cannot be continuous.

Problem 2.

If $X$ is contractable, we will eventually show that $X$ is simply connected. Below is an argument which claims to do this, but contains one or more errors. Identify them.
“Let $F$ be a homotopy between id and the constant map $X \rightarrow{x}$. Let $\alpha$ be a loop at $x$, and consider:
$$G: I \times I \rightarrow X \quad G(t, s):=F(t, \alpha(s)) .$$
This is a path homotopy from $\alpha$ to the constant map, so we conclude $\alpha$ is null-homotopic.”

Problem 3.

A collection $\mathcal{A}$ of subsets of $X$ has the countable intersection property if every countable intersection of elements of $\mathcal{A}$ is nonempty. Show that $X$ is a Lindelöf space if and only if for every collection $\mathcal{A}$ of subsets of $X$ having the countable intersection property,
$$\bigcap_{A \in \mathcal{A}} \bar{A}$$
is nonempty.