Let X = \(\mathbb{R}\) with usual topology. Then the closure of (0, 1) in \(\mathbb{R}\) equals [0, 1] as [0, 1] is the smallest closed set containing (0, 1). However, the closure of (0, 1) in the subspace topology on [0, 1) equals [0, 1) as [0, 1) is the smallest closed in the subspace topology that contains (0, 1).

Consider the subset A := [0, 1) ∪ {2} of the real line with usual topology. Note that 1 is a cluster point of A but 2 is not a cluster point of A.

What are all cluster points of {q} in VIP topology with base point p on \(\mathbb{R}\) ?

Let \((X,\Omega_i)(i=1,2)\) be two topological spaces and consider the identity function \(id\) from \(X\) onto itself. Show that \(\Omega_1\) is finer than \(\Omega_2\) iff the identity function \(id\) is continuous.

\(X\)Let X be a topological space with discrete topology and \(Y\) be any topological space. Show that any function \(f:X \rightarrow Y\) is continuous.

Let \((X_i, \Omega_i)\) be two topological spaces with topology \(\Omega_i(i=1,2)\)and let \(x_0 \in X_2\). If \(X_2\) is Hausdorff then show that for any continuous function \(f:X_1 \rightarrow X_2\), the set   \(C:= \{x \in X_1 :f(x)=x_0 \}\)is closed in \(X_1\).

If \(f:X \rightarrow Y\) is continuous and a sequence \(\{x_n\}\) in \(X\) converges to \(x \in X\), show that the sequence \(\{f(x_n)\}\) in \(Y\) converges to \(f(x)\).

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.