" When I check continuity between one of the two contacts and the brass base, it reads an open circuit."
Usually this happens with a burned out filament. If the bulb is known good, you are just not making contact with one or both probes from the continuity checker. Clean the contacts, and the base of the bulb until shiny, and try again.
When I check continuity between the two contact points, I get a closed circuit reading."
This is normal with a good bulb. The electricity is going in to one contact, through one filament, into the base, that has a connection to both filaments, and through the second filament, and out the second contact.