Existence of subbase in topology

Question

I have some question on topology.

Let $X$ be a set and $B$ is a basis of $X$. (i.e. a collection of subsets of $X$ satisfying some property of basis of $X$. Here the special property is

(1) For any $x \in X$, there is a $A \in B$ such that $x \in A$.

(2) For $A_1,A_2 \in B$, if $x \in A_1 \cap A_2$, then there is $A_3 \in B$ s.t. $x \in A_3 \subset A_1 \cap A_2$ )

Then can we always find a subbase $C$ of $X$ such that the collection of finite intersections of elements in $C$ becomes $B$?

Here, the the definition of subbase of $X$ means that the total union of it is $X$.

Thanks very much!

Answer 1

My original post was in error as you explained , No you can't get in general a sub- collection whose finite intersections give B since a collection of finite intersections from a sub=collection C is closed under finite intersections and B in general is not. For example the collection of open disks in the plane is a basis for the usual topology but a non empty intersection of disks is not a disk. However if you take any non=empty collection C and take B to be the collection of all finite nonempty intersections of sets of C then B will be a basis for a topology. This is all discussed in Kelley,General Topology ,for esample.

Answer 2

Of course.
Every topology is both a base, and a subbase for itself.
Additionally, every base is a subbase.

A collection of subsets C, is a subbase for a given topology when { $\cap$K : K finite subset of C } is a base for that topology.

A collection of subsets C with $\cup$C = X is a subbase for some topology, the topology generated by C.

Notice the significant difference.

Answer 3

Not always.

Note that the collection of finite intersections of elements of $C $ is a collection that is closed under finite intersections. So if base $B$ has not this property (and this is quite well possible for a base) then the two collections cannot coincide.

If coincidently base $B$ has the property then you can take $C=B$.