Why can the BGP protocol be implemented across routers?
But the OSPF protocol can only be implemented on directly connected routers?
Is the OSPF protocol based on IP? we know the BGP is based on TCP, so is this affect that?
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OSPF was designed as an internal gateway protocol, to chose the best path in a network under your control. It would not make sense to connect non-adjacent routers, so that feature is not part of the protocol.
BGP can be used at the edges of an autonomous system with intervening routers in the middle. So it can peer with non-adjacent routers.