In spaces where singleton points are closed, your property is equivalent to saying that the space has no isolated points. Or in other words, that it is [perfect](http://en.wikipedia.org/wiki/Perfect_space). 

Clearly, no space with an isolated point can have your property. Conversely, when singletons are closed, then you can subtract one point from any open set and thereby have a proper open subset. So if U has at least 2 points x,y, then U = U-{x} union U-{y}, giving an instance with I of size 2.