In Munkres's Topology textbook, it says,
Definition
A subbasis S for a topology on X is a collection of subsets of X whose union equals X.
The topology generated by the subbasis S is defined to be the collection T of all unions of finite intersections of elements of S.
I am having a very hard time understanding the first sentence. Let's say $X = \{1, 2, 3\}$. Then let $A = \{\{1\}, \{2\}, \{3\}\}$, or $A = \{\{1, 2\}, \{3\}\}$. Either way, $A$ is a collection of subsets of $X$, and the union of the elements in $A$ is $X$, but $A$ does not seem like a subbasis here…
What did I misunderstand here? What should be the correct interpretation of "a collection of subsets of X whose union equals X"?
Thank you in advance!
Edit:
So turned out $A$ is a subbasis in both cases, thanks @Surb for your comment.
But here is what I don't understand (which made me mistakenly think A may not even be subbasis in the first place). If $A = \{\{1\}, \{2\}, \{3\}\}$, then wouldn't "the collection T of all unions of finite intersections of elements of $A$" be {$\varnothing$}, since any intersection of any two elements in $A$ is empty? But since T is a topology, T must also contain $X$, but, in this case, it does not?
Best Answer
The first "subbase" is {{1}, {2}, {3}}. The set of all possible unions are {{1}, {2}, {3}, {1,2}, {1, 3}, {2, 3}, {1, 2, 3}}. The set of all possible intersections is {{1}, {2}, {3}, {}}.
The set of all possible unions and intersections is {{}, {1}, {2}, {3}, {1,2}, {1, 3}, {2, 3}, {1, 2, 3}}. That is a topology, in fact it contains all subsets, the "discrete topology" for {1, 2, 3}.
The second subbase is {{1, 2}, {3}}.
The set of all possible unions is {{1, 2}, {3}, {1, 2, 3}}.
The set of all possible intersections is {{1, 2}, {3}, {}}.
The set of all unions and intersections is {{}, {3}, {1, 2}, {1, 2, 3}}. Again, that is a topology for {1, 2, 3}.
If you could not see that, are you clear what a "topology" for a set is? A topology for a set, X, is a collection of subsets or X that has four properties: