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?
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?
0 commentaires:
Enregistrer un commentaire