Show that the continuous image of a compact space is again compact.

Let \(X:=\{(u,v) \in \mathbb{R^2}:- \pi \le u \le \pi ,-1 \le v \le1\}\) (Rectangle) and let \(Y:=\{(x,y,z) \in \mathbb{R^3}:x^2+y^2=1,|z| \le1\}\) (Cylinder). Verify the following:

(1) If \(u-u^{\prime} =\pm2 \pi\)then define \((u,v) \sim(u^\prime ,v^\prime)\) iff \(v=v^\prime\) . Otherwise, define \((u,v) \sim(u^\prime ,v^\prime)\) iff \((u,v) = (u^\prime ,v^\prime)\). Then \(\sim\) defines an equivalence relation on \(X\).

(2) \(f:X \rightarrow Y,f(u,v)=(cosu,sinu,v)\) is a continuous surjection. Conclude that \(X / \sim\) and \(Y\) are homeomorphic.

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.