please see the following pic:

the client initiate a half-close request and get acknowledged,if i dont misunderstand what half-close means,the client should not be able to send any data afterwards, then how can it send back an ACK later.
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It’s not the client data that is being sent, it the acknowledgement for the server data received after client does
shutdown( fd, SHUT_WR ), which means client will not send anymore, but can still receive. TheACKitself is part of the transport protocol, not the application.