Progress over the past week.
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Conversely, if $\cb \subset 2^X$ is a family such that:
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\begin{enumerate}
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\item[(TB1)] For every $x \in X$, there exists $U \in \cb$ such that $x \in U$.
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\item[(TB2)] For every $x \in X$ and $U, V \in \cb$ such that $x \in U \cap$, there exists $W \in \cb$ such that $x \in W \subset U \cap V$.
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\item[(TB2)] For every $x \in X$ and $U, V \in \cb$ such that $x \in U \cap V$, there exists $W \in \cb$ such that $x \in W \subset U \cap V$.
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\end{enumerate}
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then $\topo(\cb)$ is a topology on $X$, and $\cb$ is a base for $\topo(\cb)$.
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\end{definition}
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