Open set (equivalent definitions?)

[center o bass]

I've seen open sets $S$ of a bigger set $X$ being defined as

1) for every $x\in S$ one can find an open disk $D(x,\epsilon)$ centered at $x$ of radius $\epsilon$ such that $D$ is entirely contained in $S$. Where

$$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$
and $d$ is a metric.

2) An open set is a set that can be written as a union of open disks.

Are these two definitions equivalent in general? Or does it require $X$ to be Hausdorff. If they are in general equivalent, can you outline a proof?

 

[center o bass]

I found a proof here:
http://people.hofstra.edu/stefan_waner/diff_geom/openballs.html

 

[mklejeune]

The two definitions are equivalent if the topological space in question is metrizable.

I recommend proving for yourself that every metric space satisfies the Hausdorff axiom.

 

[WWGD]

More generally, the open subsets of a topological space $X$ are , or can be, any collection of subsets of $X$ that are closed under unions and closed under finite intersection, and the collection includes the whole space $X$ and the empty set. You then have a sub -collection of the collection of open sets that is called a basis, so that for every element $x$ is an open set $U$ , there is a basis element $B$ with $x$ contained in $B$, and $B \subset U$.