Exercise on Topology
-
- Community Team
- Posts: 426
- Joined: Mon Nov 09, 2015 1:52 pm
Exercise on Topology
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 \(\displaystyle{\mathbb{T}(f)}\) .
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 \(\displaystyle{\mathbb{T}(f)}\) .
Create an account or sign in to join the discussion
You need to be a member in order to post a reply
Create an account
Not a member? register to join our community
Members can start their own topics & subscribe to topics
It’s free and only takes a minute
Sign in
Who is online
Users browsing this forum: No registered users and 22 guests