Equivalent topology definitions Given any two absolute values on an arbitrary field k, one defines the absolute values to be equivalent if they define the same topologies. I am having trouble understanding how the following definitions of equivalent topologies T1 and T2 are the same.


*

*Every set open under T1 is open under T2

*Every T1-open ball contains a T2-open ball centered at the same point and vice versa.

*Every T1-open set contains a T2-open set, and vice versa.


1 iff 2 is clear, but I'm having trouble with 2 iff 3.
In particular, say we fix e>0 and x, and consider the ball centered at x with radius e under T1. Why can't every arbitrarily small ball centered at x under T2 not be fully contained in the set. It seems that this could still occur while an arbitrarily shaped T2- open set is contained in the ball centered at x with radius e under T1.
 A: *

*Do you ask why 2. requires a $T_2$-ball centered at the same point? You suppose that just a ball containing that center is enough? Or you suppose that just any ball jumping around $x$ is required?

If you have such a ball containing $x,$ no matter whether $x$ is its center or not, it definitely implies the existence of an open ball centered at $x.$ So, your condition is sufficient not necessary. cf. [Example 2.2, p. 232]{Analysis I, Herbert Amann}
If you have many $T_2$-balls that don't contain $x,$ what are they for? when we need one open ball centered at $x$ for all such points $x$ to show their containing set is also an open set in $T_2,$ by definition of open set. If you have many such balls but none containing $x,$ the definition is not satisfied.


*Do you ask why can't every arbitrarily small $T_2$-ball centered at $x$ be fully contained in the $T_1$-ball?

If there is one such ball, it is already sufficient, why need more? there is an upper bound of the radius (in order to have the balls contained in the set, which is to be shown to be open in $T_2$), and consequently all smaller balls are contained in the $T_1$-ball, and so if the "arbitrarily small" that you want surpasses this upper bound, it doesn't hold.
