TITLE: Are these bases for a topology?
QUESTION [5 upvotes]: I have the following topology :
$$\tau= \Bigl\{U\subseteq \mathbb{R}^2: (\forall(a,b) \in U) (\exists  \epsilon >0) \bigl([a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq U\bigr)\Bigr\}$$
Are these a basis for the previous topology:
$\beta_1= \{[a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq \Bbb R^2: (a,b)\in \Bbb R^2, \epsilon>0 \}$
$\beta_2= \{[a,a+\epsilon) \times [b-\epsilon, b+\epsilon)\subseteq \Bbb R^2: (a,b)\in \Bbb R^2, \epsilon>0 \}$
The first one is obviously a basis for $\tau$ because of the definition of $\tau$
and I would say that the second is also a basis, because $[a,a+\epsilon) \times [b-\epsilon, b+\epsilon) \subseteq [a,a+\epsilon] \times [b-\epsilon, b+\epsilon] 
$
Is it correct? what do you think?

REPLY [2 votes]: Hint. What definition of “basis” are you using?  There are a few variations of the definition around, but all the ones I know start by requiring that the sets in the basis should themselves be open sets of the topology.  It’s easy to jump straight to checking the other properties, but it’s important not to forget that first condition.

REPLY [2 votes]: Note that no element in $\beta_1$ is actually an open set in the topology $\tau$. Consider the element $A$ in $\beta_1$ with $a=b=0$ and $\epsilon=1$. The point $(1,1)\in A$ but there exists no $\epsilon>0$ such that $[1,1+\epsilon]\times[1-\epsilon,1+\epsilon]\subset A$ because $(1,1)$ is on the boundary of $A$.
For $\beta_2$, consider the same $a,b,\epsilon$ but look at the point $(0,0)$.