I've seen open sets ##S## of a bigger set ##X## being defined as(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Open set (equivalent definitions?)

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