# Whether the given Topology is Hausdorff [closed]

Consider a set $X$ and its power set ${\cal P}(X)$. Let $R$ be a non-empty subset of $X$. Define a ball of radius $R$ around a set $Z \subseteq X$ as follows:

$B(Z,R) = \{ Y \in {\cal P}(X ) | Y \triangle Z \subseteq R \}$ where $\triangle$ stands for symmetric difference of sets.

In an attempt to define a topology on ${\cal P}(X)$, call a set open if it can be written as union of balls.

My questions are:

1) Is it a valid topology?

2) If the answer to 1) is yes, then is the topology Hausdorff?

-
No, if $X$ has at least two elements, then this is not a valid topology: if $a\ne b$, then $B(Z,\{a\})\cap B(Z,\{b\}=B(Z,\emptyset)=\{Z\}$, but you did not actually make $B(Z,\emptyset)$ an open set. This also shows that the topology generated by your sets as a subbase is just the discrete topology (which is Hausdorff, but not terribly interesting). –  Emil Jeřábek Sep 28 '11 at 18:09
For future, please not that MO is intended for research-level mathematical questions, see the FAQ mathoverflow.net/faq . Basic questions like this are better suited for math.stackexchange.com . –  Emil Jeřábek Sep 28 '11 at 18:16
The problem was I did not know whether this was research level or not. Anyway, thank you. –  Spai Sep 28 '11 at 19:01
If you don't know if it's research level or not, you can safely assume it's not. –  euklid345 Sep 28 '11 at 19:05
I disagree with euklid345. One can ask a research-level question without knowing what level it is. It is quite common for professional mathematicians to be unaware of what is obvious and what is hard in distant fields. –  S. Carnahan Sep 29 '11 at 16:25
show 1 more comment

## closed as too localized by Emil Jeřábek, Andreas Blass, Captain Oates, Gjergji Zaimi, George LowtherSep 29 '11 at 0:28

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.