Monero started back synchronizing after completion or connected ! Should I just close the wallet once it completed the next time?
Monero's initial sync of a full node takes up to few days depending on your internet connectivity and also your hardware, which has to verify all synced blocks.
When you loose your internet connection, close your wallet, switch off your device, for obvious reasons your wallet will not be syncing new blocks. But after you start it again, it will resume syncing from its last position, so only blocks you missed in the meantime have to be synced with the network of nodes and verified by your wallet / daemon.
If it starts syncing from scratch after being fully synced, either your blockchain data got deleted / corrupted on your drive or it is a bug and you should report it on GitHub or discuss in the IRC / Telegram chats how to get it solved.
You can close the wallet whenever you want and it will carry on syncing from wherever it left off.