Name of the set that forms a topological space with a topology Let's assume I have a topological space $(X, \tau)$, where $X$ is a set and $\tau$ is a topology. Now I have $Y\subset X$, but $Y$ is not necessarily element of $\tau$. What do I call $X$ in my publication?
It is not the "topological space",  because that would be $(X, \tau)$, right? How can I refer to $X$? "Y is a subset of a topology imbued set"? Sounds a bit weird and "Y is a subset of a set which is used to construct a topological space..." is not exactly elegant. The same issue would be the case for a metric space. Can I say "topological space set"?
https://en.wikipedia.org/wiki/Vector_space also makes a abusive of termonology, I guess, when saying that

A vector space ... is a collection of objects called vectors, which ...


You can say that $Y$ is a subset of the underlying set of the topological space $(X, \tau)$. Usually, when a set $X$ is endowed with some "structure", you can address the set itself by calling it the underlying set. What you are doing is "forgetting" the structure and considering just the set of elements. This works for topological spaces, as well as for groups and in general whenever you have a set with operations defined between its elements.
