Topology: An Inquiry-Based Approach

Define an equivalence relation $\sim$ on ${\mathbb{R}}^2$ by $(x_1,y_1)\sim (x_2,y_2)$ whenever $x_2-x_1\in \mathbb{Z}$ and $y_2-y_1\in \mathbb{Z}\text{.}$

Let $X$ be a topological space and let $A$ be a subspace of $X\text{.}$ Define a relation $\sim$ on $X$ whose equivalence classes are $A$ and $\{x\}$ if $x\notin A\text{.}$ In this case the quotient space is denoted as $X/A$ (think of this space as obtained by crushing $A$ to a point and leaving everything else alone). Describe each of the following quotient spaces.

Let $D^2=\{(x,y)\in {\mathbb{R}}^2\mid x^2+y^2\le 1\}$ be the unit disk in ${\mathbb{R}}^2$ with the standard topology, and let $S^1=\{(x,y)\mid x^2+y^2=1\}$ be the boundary of $D^2\text{.}$ Describe the quotient spaces $D^2/\sim$ for each equivalence relation (assume that points are similar to themselves). Let $x=(s_1,t_1)$ and $y=(s_2,t_2)\text{.}$

Let $X=I\times I$ where $I$ is the interval $[0,1]$ with the standard metric topology, and define an equivalence relation on $X$ by $(s_1,t_1)\sim (s_2,t_2)$ when $t_1=t_2>0$ and $x\sim x$ for all other $x\in X\text{.}$

In the process of developing techniques of drawing in perspective, renaissance artists found it necessary to consider a point at infinity where all lines intersect. This creates a geometry that extends the concept of the real plane. This new geometry is the real projective plane $\mathbb{R}{P}^2\text{,}$ which can be thought of as the quotient space of ${\mathbb{R}}^3\setminus \{(0,0,0)\}$ with the relation ${\sim }_P$ such that $(x_1,x_2,x_3){\sim }_P(y_1,y_2,y_3)$ in ${\mathbb{R}}^3\setminus \{(0,0,0)\}$ if and only if there is a nonzero real number $k$ such that $y_1=k{x}_1\text{,}$ $y_2=k{x}_2\text{,}$ and $y_3=k{x}_3\text{.}$ In the projective plane, parallel lines intersect at a point at infinity, just as they seem to with our human vision.

Let $X={\mathbb{R}}^2$ with the standard topology, and let $Y=\{x\in \mathbb{R}\mid x\ge 0\}$ with the standard topology. Let $f:X\to Y$ be defined by $f((x,y))=x^2+y^2\text{.}$

Let $(X,\tau )$ be a topological space. A subspace $A$ of $X$ is a retract of $X$ (or that $X$ retracts onto $A$) if there is a continuous function $r:X\to A$ such that $r(a)=a$ for all $a\in A\text{.}$ Such a map $r$ is called a retraction. Intuitively, a subspace $A$ of $X$ is a retract of $X$ if we can continually collapse (or retract) $X$ onto $A$ without moving any of the points in $A\text{.}$ Certain types of retracts, namely deformation retracts, are important in algebraic topology.

For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.