One of my neighbours died not long ago. I popped a condolence card through the letterbox, and have chatted to his widow when we have met.
I wasn’t sure what to do about a Christmas card, not wanting to miss her out, but knowing that a “have a happy Christmas” would not be appropriate. I found a very nice card saying “with sympathy at Christmas time” and a nice message inside saying that we were thinking of them at this difficult time. People often don’t know how to interact with the bereaved, as I know from experience. I sometimes make a point of mentioning my late husband in conversation, so that people know that it’s ok to talk about him.