Definition of a Basis for a Topology – Intersection of Basis Elements and Possibility of being a Topology

Question

Part of the definition of a basis for a topology for a set $X$ states:

If $x$ belongs to the intersection of two basis elements $B_1$ and $B_2$, then there is a basis element $B_3$ containing $x$ such that $B_3 \subset B_1 \cap B_2$.

I would like to clarify that this definition does not rule out the possibility that $B_3=B_2$ or $B_3=B_1$. Because applying this definition to, for instance, $B_1$ and $B_3$ we should find a set $B_4$ containing $x$ such that $B_4 \subset B_1 \cap B_3$.

I have another question about basis. Is it possible for a basis on $X$ to be also a topology on $X$? For instance for set $X=\{1,2\}$, basis $\mathcal{B} = \{ \phi, \{1\}, \{1,2\}\}$ is also a topology (I hope I am applying the definitions correctly). If $\mathcal{B}$ it is indeed a basis and topology for this particular example, then I find this a bit counter-intuitive because one expects a basis to be smaller (properly contained) than a topology.

Answer 1

As pointed out in the comment, every topology is its own basis. But every basis need not be a topology.

The proof of every topology being a basis is quite simple.

1. For every element $x\in X$, we have an element in the topology which contains $x$. This element is $X$ itself.
2. For two open sets $X_1,X_2,X_1\cap X_2$ is open and belongs to the topology as well.

Thus both properties of a basis are satisfied.