Is the Definition of a Subbasis in Munkres' Topology Textbook Flawed?

[ak416]

in the munkres book, they define A to be a subbasis of X if it is a collection of subsets of X whose union equals X. They define T, the topology generated by the subbasis to be the collection of all unions of finite intersections of elements of A.
This definition seems to be flawed because, given that definition, i can easily construct a set A that is a subbasis but won't generate a topology. For example, given any set X let A be a partition on X (A is made of disjoint sets). Any finite intersection here would be empty and therefore T (the topology generated by A) would be empty. I think the only way to resolve this issue is to add that X must be an element of A.
I think they assume this but it just bothers me that they didnt write it down explicitly.

 

[ak416]

Ok i think i have resolved this problem. If you consider each element of A to be an intersection with itself, then it will work...

 

[matt grime]

Since, by definition a topology is a set closed under finite intersection and arbitrary union, then your definition cannot be flawed, and your second post correctly identifies your problem.