Exercise:
Suppose $\mathcal{S}$ is a collection of subsets of $X$ whose union equals $X$. Prove there is a coarsest topology $\mathcal{T}$ containing $\mathcal{S}$ and that the collection of all finite intersections of sets in $\mathcal{S}$ is a basis for $\mathcal{T}$. In this situation, the collection $\mathcal{S}$ is called a subbasis for the topology $\mathcal{T}$.
My attempt:
Given all topologies $\mathcal T_{\alpha}$ on $X$ such that $\mathcal S \subseteq \mathcal T_{\alpha}$ (where we have $\alpha \in \mathsf A$, using $\mathsf A$ as an index set to avoid assumptions of countability), define $\mathcal T := \bigcap_{\alpha \in \mathsf A} \mathcal T_{\alpha}$. I think it is clear that $\mathcal S \subseteq \mathcal T$, and if $\mathcal T'$ is any other topology containing $\mathcal S$, then have $\mathcal T \subseteq \mathcal T'$. It is necessary to prove that $\mathcal T$ is in fact a topology, but I found this fairly straightforward and in the interest of brevity I will omit the proof.
The last thing to do is prove that $\mathcal B := \{ \text{all finite intersections of } S \in \mathcal S \}$ is a basis for $\mathcal T$. I use the definition of basis given on page 2, shown below.
Definition 0.2. A collection $\mathcal{B}$ of subsets of $X$ is a basis for a topology on $X$ if and only if
(i) For each $x \in X$ there is a $B ∈ \mathcal{B}$ such that $x ∈ B$.
(ii) If $x ∈ A \cap B$ where $A, B \in \mathcal{B}$, then there is at least one $C \in \mathcal{B}$ such that $x \in C \subseteq A \cap B$.
(i) For any $x \in X$, we know there is some $S \in \mathcal S$ such that $x \in S$. Since $S$ itself is a degenerate intersection we see that $S \in \mathcal B$, therefore the first condition is satisfied.
(ii) If $x \in A \cap B$ where $A, B \in \mathcal B$, then we know that $A = \bigcap_{i = 1}^m S_i$ and $B = \bigcap_{i = 1}^n S_i'$. Clearly $A \cap B$ is another finite intersection, hence $A \cap B \in \mathcal B$.
Questions and comments:
- I think this exercise is relatively straightforward, but my inexperience with the definitions causes some concern that I've made some elementary error in my proof.
- Intuitively a basis is something that can give rise to a topology through unions, and a subbasis is something that can give rise to a basis through finite intersection. Right?
I appreciate any feedback.
