I am trying to understand what is the difference between TCP Half Open Connection and TCP Half closed connection can any one tell what exactly are they?
This post expands on half closed connections. For half open connections see KContreau's correct description.
What are Half Closed Connections? Or: It's Not a Bug--it's a Feature!
Every TCP connection consists of two half-connection which are closed independently of each other. So if one end sends a FIN, then the other end is free to just ACK that FIN (instead of FIN+ACK-ing it), which signals the FIN-sending end that it still has data to send. So both ends end up in a stable data transfer state other than ESTABLISHED--namely FIN_WAIT_2 (for the receiving end) and CLOSE_WAIT (for the sending end). Such a connection is said to be half closed and TCP is actually designed to support those scenarios, so half closed connections is a TCP feature.
The History of Half Closed Connection
While RFC 793 only describes the raw mechanism without even mentioning the term "half closed", RFC 1122 elaborates on that matter in section 220.127.116.11. You may wonder who the hell needs that feature. The designers of TCP also implemented the TCP/IP for the Unix system and, like every Unix user, loved I/O redirection. According to W. Stevens (TCP/IP illustrated, Section 18.5) the desire to I/O-redirect TCP streams was the motivation to introduce the feature. It allows the FIN ack take the role of or be translated as EOF. So it's basically a feature that allows you to casually create impomptu request/response-style interaction on the application layer, where the FIN signals "end of request".
When TCP establishes a connection, it is considered guaranteed since there is a handshake that takes place:
At that point the connection is established, and data begins to flow. In contrast, a UDP packet is not guaranteed, and is just sent in the hopes it gets there.
Officially, according to the RFC's, a half-open TCP connection is when one side of the established connection has crashed, and did not send notification that the connection was ending. This is not the common usage today.
Unofficially, if can refer to an embryonic connection, which is a connection in the process of being established.
Half-closed is the opposite of that unofficial definition. It is a state somewhere in the middle where the computers are tearing down the established connection.