Is the definition of subbasis well-defined?

Question

There are more definitions of subbasis in general topology. I chose the one that seams me to be the simplest:

Definition. Let $(X,\tau)$ be a topological space. A subbasis of $\tau$ is a subset $B\subseteq\tau$ that generates the topology T. This means that $\tau$ is the smallest topology containing $B$ (any topology $\tau'$ on $X$ containing $B$ must also contain $\tau$).

Note. From my understanding such a definition can be made only after proving that the smallest topology containing $B$ is something well-defined and unique. I mean: not all topologies containing $B$ have to be comparable. We can consider $S:=\{\tau:B\subseteq \tau \text{ and $\tau$ is a topology of $X$}\}$, but then we have to prove that $S$ has a minimum.

How can I formulate a Lemma that legitimates the definition above? Or, if such a lemma is not needed, why is it not needed?

Why in topology the word collection of elements is so often used instead of subset?

Answer 1

Lemma: Let $X$ be a set and let $S$ be a subset of $\mathcal{P}(X)$. Then, among all topologies in $X$ there is one and only one topology $\tau$ such that:

1. $\tau\supset S$;
2. if $\tau^\star$ is a topology in $X$ and $\tau^\star\supset B$, then $\tau^\star\supset\tau$.

Proof: Let $\mathcal T$ be the set of all topologies in $X$ which contain $B$. Then $\mathcal{T}\neq\emptyset$, since $\mathcal{P}(X)\in\mathcal T$. Let $\tau=\bigcap_{T\in\mathcal T}T$. Then $\tau$ is a topology and $\tau\supset B$. Besides, it follows from the definition of $\tau$ that, if $\tau^\star$ is a topology in $X$ and if $\tau^\star\supset B$, then $\tau^\star\supset\tau$.

Answer 2

Hint: For any set of topologies $\{\tau_I\}$, their intersection $\bigcap_I \tau_I$ is also a topology.