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 ...

Solution 1:

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.

Solution 2:

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.