A set can be neither open nor closed or both open and closed. In a discrete topology, every set $A\subset\mathcal{S}$ is both open and closed, whereas, in a trivial topology, any set $A\neq\emptyset$ or $\mathcal{S}$ is neither open nor closed.
The link is here on page 2
I could understand in the discrete topology, every subset of $\mathcal{S}$ are in the topology so they are open by definition. since their complement is also in the topology,which means the complements are open so they are also closed.
Questions
- Does $\emptyset$ and $\mathcal{S}$ both open and closed in trivial topology? I find both are open under trivial topology on the wiki, I think they are closed as well like the above reasoning.
- In trivial topology, I don't know why the set itself $\mathcal{S}$ is neither open nor closed in the notes.
Please help me out with these two questions, thank you so much.
Best Answer
The empty set and the whole set are always open, by definition of a topology. Being the complement of each other they are always closed.
In the notes is said that any subset $A$, $A \neq S$, $A \neq \emptyset$ is neither oper nor closed for the trivial topology.