Let \(E\) be a non-empty proper subset of the topological space \(X\). Consider the characteristic function \(\chi _E\) of \(E\). Show that \(\chi _E\) is continuous iff \(E\) is a closed and open subset of \(X\).

Consider topological spaces \(X_i=(X,\Omega_i),i=1,2\) such that \(\Omega_1\) is finer than \(\Omega_2\). Show that \(f:X_1 \rightarrow Y\) is continuous if so is\(f:X_2 \rightarrow Y\).

Let \(X\) be a topological space and \(Y\) be a topological space with ordered topology. Let \(f,g:X \rightarrow Y\) be continuous functions. Show that the set \(U:=\{x \in X:f(x)>g(x) \}\) is open in \(X\).

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 \(Y= \mathbb{R^n}\)