Prove: Composition of continuous functions is continuous.

Let \(X:=[0,1]\) and \(A_0:=\{0\} , A_n:=[1/n,1]\). Then the function \(f:[0,1] \rightarrow \mathbb{R}\)given by \(f(0)=1,f(x)=1/x(0<x\le1)\) is discontinuous at 0. However, \(f|_{A_n}\) is continuous for every \(n \ge 0\).

Show that the function \(g: (0,1) \rightarrow \mathbb{R}\) given below is continuous on irrationals and discontinuous on rationals:

\(g(x) = \frac{1}{q} \; if \; x \in \mathbb{Q} \; \cap \; (0,1)\)and \(x = \frac{p}{q}\) in reduced

form = 0 otherwise.

We say that a set is a \(G_{\delta}\) set if it is countable intersection of open sets.

Show that a bijective continuous map from a compact metric space into a metric space sends closed sets to closed sets, and hence it is a homeomorphism.

Define \(f: \mathbb{R} \rightarrow [0,1]\) by \(f(x) =x\) if \(|x| \le 1,\) and \(f(x)= \frac{1}{|x|}\) if \(|x| \ge1\). Then f is continuous on \(\mathbb{R}\). Note that \(f\) is onto but not one-one.

 Show that the interval \((a,b) \subseteq \mathbb{R}\) is homeomorphic to any other interval \((c,d) \subseteq \mathbb{R}\).

Show that \(e^{-x}\) is a homeomorphism from \((0, \infty)\) onto \((0,1)\).

Which of spaces \(X\) and \(Y\) are homeomorphic:

(1) \(X= \mathbb{R}\) and \(Y=[0,1)\)

(2) \(X=\mathbb{R}\) and \(Y=[0,1]\)

(3) \(X=[1, \infty)\) and \(Y=(0,1]\)

(4) \(X=\mathbb{R}\) and \(Y=(0,1)\)

(5) \(X=\mathbb{Q}\) and \(Y=\mathbb{Q}\)

(6) \(X=\{(x,y,z) \in \mathbb{R^3} : x^2+y^2 =z ,r<z<R \}\) and \(Y =A(0,r,R)\)

(7) \(X= \{x \in \mathbb{R^{n+1}}:x_{n+1}=0 \}\) and \(Y= \mathbb{R^n}\)

Verify that \(\phi(t)=e^{it}\) is a homeomorphism between \((0,2\pi)\) and \(\mathbb{T} \backslash \{1\}\), where \(\mathbb{T}\) denotes the unit circle in the complex plane. Conclude that unit circle minus a point is homeomorphic to the real line.

Show that \(X=A(0,r,R)\) (open annulus) and \(Y= \mathbb{T}\)(unit circle) are not homeomorphic

Let \(X=S^n \backslash \{(0,\ldots ,1)\} \subseteq \mathbb{R^{n+1}}\) and \(Y= \mathbb{R^n}\) , and define

\(f:X \rightarrow Y\) and \(g:Y \rightarrow X\) by

\(f(x_1, \ldots ,x_{n+1}) =\large( \frac{x_1}{1-x_{n+1}}, \ldots , \frac{x_n}{1-x_{n+1}}),\\g(x_1, \ldots ,x_n)=\large(\frac{2x_1}{{||x||_2}^2+1} ,\ldots ,\frac{2x_n}{{||x||_2}^2+1} , \frac{{||x||_2}^2-1}{{||x||_2}^2+1} )\)

Verify that \(f\) and \(g\) are homeomorphisms such that \(f^{-1}=g\).