Trieste surnames: Redemont and Bus?

[annettepr]

Hello, I just found out that one of my ancestors was born in Trieste, Italy, and migrated to Puerto Rico around 1817. In his naturalization records it is stated that his parents' names were Pedro Bus (in other records it is written Buz) and Juana Redemont. Are these Trieste surnames? Typically foreign surnames were misspelled because the transcriber wrote down the phonetical sounds or what he thought he heard, so I have a suspicion the correct spelling might be another and it is crucial to get the names right to further my research. Thanks in advance for any help.