Bases for a topology

Question

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:

$\beta_1= \{[a,a+\epsilon] \times [b-\epsilon, b+\epsilon]\subseteq \Bbb R^2: (a,b)\in \Bbb R^2, \epsilon>0 \}$

$\beta_2= \{[a,a+\epsilon) \times [b-\epsilon, b+\epsilon)\subseteq \Bbb R^2: (a,b)\in \Bbb R^2, \epsilon>0 \}$

I have asked this question before here:https://math.stackexchange.com/posts/536105/edit

And I got stuck for $\beta_2$, considering e $a,b=0,\epsilon=1$ this set is not open in our topology. They gave me the advice to look at the point $(0,0)$.What is the problem here?

Thank you

You mean $\epsilon=1$ don't you? – Stefan Hamcke Oct 22 '13 at 23:21 — Stefan Hamcke, Oct 22 '13 at 23:21
yes, sorry. I take a=b=0 and $\epsilon=1$ – Blanca Oct 22 '13 at 23:23 — Blanca, Oct 22 '13 at 23:23

score 0 · Answer 1 · answered Oct 23 '13 at 00:46

I'm using the following definition:

A collection $\beta$ is a basis for a given topology $\eta$ over a set $X$ iff $$(\forall U\in\eta)(U=\bigcup\left\{B:B\in\beta,B\subseteq U\right\})$$

Notice that I don't assume that $\beta\subseteq\eta$ in this definition.

You should already know (if not, prove this) that a collection $\beta$ is a basis for a given topology $\eta$ (over a set $X$) iff the following holds: given $U\subseteq X$, $$U\in\eta \iff (\forall x\in U)(\exists A\in\beta)(x\in A\subseteq U)\qquad (1)$$

Using that, it should be easy for you to show (using $\epsilon/2$ arguments) that $\beta_2$ is a basis for $\tau$.

Bases for a topology

1 Answers1