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 ...


Well, it depends on the context in which you are relating $X$ and $Y$.
Usually, in the definition of a topology on a set $X$, the pair $(X, \tau)$ is formally called a topological space. However, if there is no confusion (since a set can be given more than one topology) you can "abuse" a little the language and refer to $X$ as a topological space.
If you want to remark both that $Y \subset X$ and that $(X, \tau)$ is a topological space, then I totally agree with  the comment @Neal made above.
