Given a topological space $(X,\cal T)$ a subcollection $\cal A$ of $\cal T$ is a base of open sets for $\cal T$ if any open set $O$ of $\cal T$ is union of any subcollection $\mathcal A_O$ of $\cal A$ so that it is usuasl to say that a collection $\mathcal S_O$ is a subbase of open sets if the collection
$$
\mathcal F_O:=\Big\{Y\in \mathcal P(X):Y=\bigcap \mathcal A_n\,\text{where }\mathcal A_n\subseteq\mathcal A\,\text{is such that }|\mathcal A_n|=n\,\text{for }n\in\omega\Big\}
$$
of finite intersection of $\mathcal S_O$ is a base of open sets.
However it is a well know result that a topology can be defined using closed sets with interchanging finite intersection with arbitrary intersection and arbitrary union with finite union so that a collection $\cal C$ of $\cal T$ is a base of closed sets for $\cal T$ if any closed set $C$ is intersection of a subcollection $\cal C_C$ of $\cal C$ (see here for details) so that I would like to know if it is possibile to define a subbase $\cal S$ of closed sets if the collection
$$
\mathcal F_C:=\Big\{Y\in \mathcal P(X):Y=\bigcup \mathcal C_n\,\text{where }\mathcal C_n\subseteq\mathcal C\,\text{is such that }|\mathcal C_n|=n\,\text{for }n\in\omega\Big\}
$$
of finite union of $\mathcal S_C$ is a base of closed sets: indeed, it is a well know resul that if $\mathcal A_O$ and $\mathcal C_C$ are base of open and closed sets then the collections
$$
\mathcal A_O^*:=\{X\setminus A_O:A_O\in\mathcal A\}\quad\text{and}\quad\mathcal C^*_C:=\{X\setminus C_C:C_C\in\mathcal C_C\}
$$
are a base of closed and open sets respectively so that I thought that by symmetry it is possibile to define a subbase of closed sets obtaining analogous results for open and closed subbases as the following observation shows.
So we observe that if $\mathcal F_O$ is a subbase of open sets then for any closed set $C$ the identity
$$
X\setminus C=\bigcup_{i\in I}\Biggl(\bigcap_{h_i\in n_i}(F_O)_{h_i}\Biggl)
$$
holds where $n_i\in\omega$ and $(F_O)_{h_i}\in\mathcal F_O$ for any $h_i\in n_i$: thus observing that
$$
C=X\setminus\bigcup_{i\in I}\Biggl(\bigcap_{h_i\in n_i}(F_O)_{h_i}\Biggl)=\bigcap_{i\in I}\Biggl(X\setminus\bigcap_{h_i\in n_i}(F_O)_{h_i}\Bigg)=\bigcap_{i\in I}\Biggl(\bigcup_{h_i\in n_i}\Big(X\setminus(F_O)_{h_i}\Big)\Biggl)
$$
we conclude that the collection
$$
\mathcal F_O^*:=\{X\setminus F_O:F_O\in\mathcal F_O\}
$$
is a subbase of closed sets. Moreover by analogous arguments it is possible to prove that if $\mathcal F_C$ is a subbase of closed set then the collection
$$
\mathcal F_C^*:=\{X\setminus F_C:F_C\in\mathcal F_C\}
$$
is a subbase for open sets.
So with respect this facts could it make sense define a subbase for closed sets? Could someone help me, please?
Best Answer
There is no problem to define (sub)base for the closed sets, just use De Morgan's laws.
In a topological space $(X,\mathcal{O})$ let $\mathcal{F}$ denote the family of closed sets A subfamily $\mathcal{B}$ of $\mathcal{F}$ is a base for the closed sets if every member of $\mathcal{F}$ is the intersection of a subfamily of $\mathcal{B}$. A subfamily $\mathcal{S}$ of $\mathcal{F}$ is a subbase for the closed sets if the family $\mathcal{S}^+$ of finite unions of members of $\mathcal{S}$ is a base for the closed sets; in extreme cases you have to add $\emptyset$ and $X$ explicitly to $\mathcal{S}^+$.
An alternative definition/characterization: $\mathcal{S}$ is a subbase for the closed sets of $(X,\mathcal{O})$ if $\mathcal{O}$ is the smallest topology whose family of closed sets contains $\mathcal{S}$.