Synthetic Basis formed from Synthetic Sub-Basis
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Theorem
Let $X$ be a set.
Let $\SS$ be a synthetic sub-basis on $X$.
Define:
- $\ds \BB = \set {\bigcap \FF: \FF \subseteq \SS, \text{$\FF$ is finite} }$
Then $\BB$ is a synthetic basis on $X$.
Proof
We consider $X$ as the universe.
Thus, in accordance with Intersection of Empty Set, we take the convention that:
- $\ds \bigcap \O = X \in \BB$
By Set is Subset of Union: General Result, it follows that:
- $\ds X \subseteq \bigcup \BB$
That is, axiom $(\text B 1)$ for a synthetic basis is satisfied.
We have that $\BB \subseteq \powerset X$.
Let $B_1, B_2 \in \BB$.
Then there exist finite $\FF_1, \FF_2 \subseteq \SS$ such that:
- $\ds B_1 = \bigcap \FF_1$
- $\ds B_2 = \bigcap \FF_2$
It follows that:
- $\ds B_1 \cap B_2 = \bigcap \paren {\FF_1 \cup \FF_2}$
proof that $\ds \bigcap_{i \mathop \in I} \bigcap \mathbb S_i$ is equal to $\ds \bigcap \bigcup_{i \mathop \in I} \mathbb S_i$
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By Union is Smallest Superset, $\FF_1 \cup \FF_2 \subseteq \SS$.
We have that $\FF_1 \cup \FF_2$ is finite.
Hence $B_1 \cap B_2 \in \BB$, so it follows by definition that axiom $(\text B 2)$ for a synthetic basis is satisfied.
$\blacksquare$
Note
Note that by this construction, any collection of subsets of $X$ can form a synthetic basis and thus generate a topology on $X$.
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction