The nostrils are connected to the throat through the sinuses as are the ear canals. Topologically making them all part of the same mouth-anus throughole.
For the more formal definitions of topological holes, look up Betti numbers, I think there's a VSauce video about it? For humans, the number of holes is at least 3 (but I'm sure I'm missing many more holes). Unless you're missing the entire septum.