Answer Question
No, you should not let your neighbor's cat inside your house. This will just create a bad tendency for the cat to want to come inside your house more often. This could create problems with the cat and your neighbors, if they find out you are letting their cat in your house. If the cat appears to be hungry then you should only feed the cat outside (not inside your house). You can just leave a little cat food in a dish outside for the cat to eat.
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