Does the complement of a set being closed necessarily imply that the set itself is open? Could the set be both/neither open and/nor closed if its complement is strictly closed?
[Math] Open and Closed Sets definition
general-topology
general-topology
Does the complement of a set being closed necessarily imply that the set itself is open? Could the set be both/neither open and/nor closed if its complement is strictly closed?
Best Answer
Definition. A set is closed if and only if its complement is open.
In every topological space $(X,\tau)$ there are at least two set which are both open and closed: $\varnothing$ and $X$.
If the discrete topology, where $\tau={\mathcal P}(X)$, every subset of $X$ is both open and closed.
A topological space is connected if and only if the only subsets which are both open and closed are $\varnothing$ and $X$.
In $\mathbb R$, there are $\boldsymbol{c}=2^{\aleph_0}$ open subsets and as many closed, while there are $2^{\boldsymbol{c}}$ subsets which are neither open nor closed.