I don't know of any performance penalties in frequent closing/opening of the database (regardless of its size). I think the answer to this question also depends on what type of application is accessing the database.
Do you "re-query" the database a lot?
Then it seems rectified to keep it open.
Do you fetch different data each time you fetch something?
Again, it seems reasonable to leave it open (as you won't gain in caching the data instead).
Are there any other applications accessing the same database?
If there is a risk for concurrency or blocking issues, it might be wise to close the database after finished reading/writing from/to it.
Generally I would say that you might gain more in caching data than in leaving the database open (contra closing it) when optimizing for performance.
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