Synthetic basis and analytic basis Definition of "Synthetic Basis":
Let $X$ be a set!
Let $\mathscr{B}$ be a subset of $P({X})$ satisfying below two conditions:
(i) $X\subset \bigcup \mathscr{B}$
(ii) $\forall A,B\in \mathscr{B}, \exists \mathscr{A}\subset \mathscr{B}, \bigcup \mathscr{A}=A\cap B$
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Question1:
Let $X$ be a set and $\mathscr{B}_1$ and $\mathscr{B}_2$ be distinct synthetic bases for $X$.
I have proven that $\tau_i\triangleq \{\bigcup \mathscr{A} \subset X|\mathscr{A} \subset\mathscr{B}_i\}$ are topologies on $X$.
I believe $\tau_1$ and $\tau_2$ are not necessarily the same, but i'm not sure. Is there any simple example?
Question 2:
Munkres, Topology p.80 : Let $X$ be a set and $\mathscr{B}$ be a basis for a topology $\tau$ on $X$. Then $\tau$ equals the collection of all unions of elements of $\mathscr{B}$.


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*I don't get this statement. If the author meant "analytic basis", then it's just the definition, that is, it should not be a theorem. So i guess he meant 
"Synthetic basis". However, if he really meant that is this theorem true? How?

 A: Here are two examples of synthetic bases on $\mathbb{R}$ which generate different topologies:


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*$\mathscr{B}_1 = \{ ( a , b ) : a < b \in \mathbb{R} \}$.

*$\mathscr{B}_2 = \{ [ a , b ) : a < b \in \mathbb{R} \}$.

*$\mathscr{B}_3 = \{ \{ a \} : a \in \mathbb{R} \}$.


The topology generated by $\mathscr{B}_1$ is the usual (Euclidean metric) topology on $\mathbb{R}$.  The topology generated by $\mathscr{B}_2$ is the Sorgenfrey topology (or lower-limit topology).  The topology generated by $\mathscr{B}_3$ is the discrete topology on $\mathbb{R}$.
There are simpler examples (since any topology is also a base for itself, and so any two distinct topologies on the same set will be bases for different topologies).

Munkres seems to define the topology generated by a base $\mathscr{B}$ slightly differently, opting instead to define it to be $$\tau = \{ U \subseteq X : ( \forall x \in U ) ( \exists B \in \mathscr{B} ) ( x \in B \subseteq U ) \}.$$  This is easily seen to be equivalent to the definition you have given, and the Theorem you mention is essentially just a formal statement of this fact.
