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Let \(\displaystyle{\left(X,\mathbb{T}\right)}\) be a topological space and \(\displaystyle{f:X\longrightarrow X}\) a function.

Consider \(\displaystyle{\mathbb{T}(f)=\left\{A\in\mathbb{P}(X): f(A)\subseteq A^{0}\right\}}\) .

1. Prove that \(\displaystyle{\mathbb{T}(f)}\) is a topology on \(\displaystyle{X}\) .

2. If \(\displaystyle{f=Id_{X}}\), then \(\displaystyle{\mathbb{T}(f)=\mathbb{T}}\) .

3. If \(\displaystyle{f(x)=x_{0}\,,x\in X}\), then find \(\displaystyle{\mathbb{T}(f)}\) .

4. Prove that if \(\displaystyle{f\circ f= Id_{X}}\), then \(\displaystyle{\mathbb{T}(f)\subseteq \mathbb{T}}\) .

5. We define \(\displaystyle{f:X\longrightarrow X}\) by


\(\displaystyle{f(x)=\begin{cases}

x\,\,\,\,\,\,\,\,,x\in X-\left\{x_1\,,x_2\right\}\\

x_1\,\,\,\,\,,x=x_2\\

x_2\,\,\,\,\,,x=x_1

\end{cases}}\)



where \(\displaystyle{x_1\,,x_2\in X\,,x_1\neq x_2}\). Find \(\displaystyle{\mathbb{T}(f)}\) .