This is such an important conversation.
In the sort of small, face-to-face communities of Africa where I've lived and researched childhood, this would be a non-starter question because there are always dozens (or even hundreds) of adults and teenagers within view of any child, and nothing disastrous could happen to a 'child left alone' because, in effect, no child is ever alone.
In our contemporary, urban societies, that's far from the case. Urban life has great disadvantages, but you've just hit on one of the major ones: the separation/loneliness factor. Combating the lack of community that characterizes modern life in cities requires major efforts. But it could be worth the effort!
Can you try and find some neighbours with children, to start a cooperative agreement? You can each be available as 'back-up' if need be. Your ten-year-old is also old enough to be able to call such a neighbour, in case of emergency.
How about constructing a list of several reliable neighbours, with their home and mobile phone #s, to leave with your child whenever you leave him alone?
If you don't know any such neighbours--this is a great time to try and find some. Post a notice in a lift? Approach another mum out walking with a pram? Start a neighborhood list-serv?