Let $X:=[0,1] \times[0,1]$ (Unit Square) and let $Y= \mathbb{T} \times \mathbb{T}$ (Torus). Verify the following:

(1) If $s_1-s_2= \pm1$ then define $(s_1,t_1) \sim (s_2,t_2)$ iff $t_1=t_2$. If $t_1-t_2 = \pm1$ then define $(s_1,t_1) \sim (s_2,t_2)$ iff $s_1=s_2$. If $(s_1,t_1),(s_2,t_2) \in [0,1] \times [0,1]$ then $(s_1,t_1) \sim (s_2,t_2)$ provided $s_1=s_2$ and $t_1=t_2$. Then $\sim$ defines an equivalence relation on $X$.

(2) $f:X \rightarrow Y$ given by $f(s,t)=(e^{2\pi is},e^{2\pi it})$ is a continuous surjection. Conclude that $X / \sim$ and $Y$ are homeomorphic.

Let $X:=[0,1] \times [0,1]$. Define the equivalence relation $\sim$ on $X$ as follows: $(x,y) \sim (x^\prime,y^\prime)$ iff either $(x=0,x^\prime=1$and $y=1-y^\prime)$ or $(x=x^\prime$ and $y=y^\prime)$. The quotient space is known as the Mobius strip  .

Define the equivalence relation $\sim$ on $X$as follows:

$(x,y) \sim (x^\prime,y^\prime)$ iff either $(x=0,x^\prime=1)$ or $(x=1-x^\prime,y=0,y^\prime=1)$ or $(x=x^\prime$and $y=y^\prime)$. The quotient space is known as the Klein’s bottle.

 Let $X:=\overline{\mathbb{D}} \subseteq \mathbb{R^2}$ (Closed Unit Disc) and let $Y:=\mathbb{S^2} \subseteq \mathbb{R^3}$ (Unit Sphere). Verify the following:

(1) If $z,w \in \mathbb{T}$then set $z \in w$. If $z,w \in \mathbb{D}$ then $z \in w$ provided $z=w$. Then $\sim$ defines an equivalence relation on $X$.

(2) $f:X \rightarrow Y,f(tx,ty)=(cos\pi(1-t),xsin\pi(1-t),ysin\pi(1-t))$ for $0 \le t \le 1,(x,y) \in \mathbb{T}$ is a continuous surjection.

Conclude that $X/ \sim$ and $Y$ are homeomorphic.

Find one-point compactifications of $\mathbb{R}$ and $\mathbb{R^2}$ .

Let $\mathbb{R^{n+1}} \backslash \{0\}$ be the punctured Euclidean space and let $\mathbb{S^n}$ denote the unit sphere in $\mathbb{R^{n+1}}$. Define $g : \mathbb{R^{n+1}} \backslash \{0\} \rightarrow \mathbb{S^{n}}$ by $g(x)=x \backslash ||x||_2$. Show that $g$ is an open map.

Show that $\mathbb{P^1}(\mathbb{R})$ is homeomorphic to the unit circle $\mathbb{T}$.

The space $\mathbb{Q}$ is not connected. In fact, for any irrational $x \in \mathbb{R}$, the set $[-x,x] \cap \mathbb{Q}= (-x,x) \cap \mathbb{Q}$ is both open and closed in the subspace topology on $\mathbb{Q}$.

If $A$ is connected and $B$ is a set such that $A \subseteq B \subseteq \overline{A}$ then $B$ is also connected.

 Show that the complement of any countable subset $C$ in $\mathbb{R^2}$ is path-connected.

 Show that the continuous image of a path-connected space is path-connected.

For open subset $U$ of $\mathbb{R^n}$ , show that $U$ is connected if and only if $U$ is path-connected.