After closing a StreamWriter the `StreamReaderProtocol.connection_lost` on the other end is not getting called. In this case the StreamReader is at EOF but calling write/drain does not raise any Exception (and sending data to Nirvana). 

I would expect that StreamWriter.is_closing returns True after the close and calling write/drain raises immediately and not just after the second call. Please see attached example. I see the same behavior with Proactor and Selector event loop on Windows.

Maybe this is expected behavior. But in this case it is completely undocumented. Should there be a check for `StreamReader.at_eof` (and maybe `StreamReader.exception`) before writing to the StreamWriter?

This might be related to bpo-34176.

After having a closer look I fear that there isn't a correct implementation for half closed sockets and returning True from eof_received results in a fundamentally broken state machine.

I'm not sure if a selector implementation can handle half closed sockets, at least I'm pretty sure that this is not supported on Windows (https://docs.microsoft.com/en-us/windows/desktop/api/winsock2/nf-winsock2-select).

I think that this issue is just for Windows right?

I add Windows to required a Windows' dev.

No, I'm seeing the same issue on MacOS. Attached modified example.

Ok, I am working on that.