I know this isn't really that important in grand scheme of things, but anyways: I'm taking topology in college rn, and we defined neighbourhood to be a set N, subset of ambient space X, such that there is an open set U containing x, such that U is subset of N.

Therefore, non open sets can also be neighbourhoods, but they are "useless" in the sense that firstly, basically every single definition and theorem involving term "neighbourhood" is equivalent to version of that statement where "neighbourhood" is swapped with "open neighbourhood", and secondly, just in general when we are working with non open neighbourhood N, we are ultimately interested in finding that open set U that is in "sandwich" between x and N, i.e. we are looking for the open neighbourhood anyway. So why not define a neighbourhood of x to be any *open* set containing x?

My professor said that indeed they are basically pointless, but purely for traditional reasons the definition remains as such. Wonder if you all thought the same?