How does one interpret functions of topological spaces?

Question

Let $f: X \to Y$ be a map of sets. We are given that $X$ is a topological space. We are to show that there is a topology on $Y$ making $f$ continuous, and moreover, determine if this topology is unique.

Should I read it as "$X$ is just a set on which there is given a topology, i.e sets that are open in $X$ are predetermined" or "$X$ is a collection of subsets that form a topological space"?

I am not asking for assistance on the exercise itself.

Answer 1

For many sorts of structures, it is common to use the same letter to refer both to the structure and its 'underlying set'.

So if $T$ is a topology on $X$, we would commonly write $X$ when we mean the topological space $(X,T)$, when there is no ambiguity. (if we were considering several different topologies on $X$, we wouldn't do this)

A more familiar example is that we use the symbol $\mathbb{R}$ to denote the set of real numbers, the field of real numbers, the topological space we call the real line, the metric space we call the real line, the additive abelian group of real numbers, the ....

Answer 2

Should I read it as "is just a set on which there is given a topology.."

Yes.

The topology induced in $Y$ by $f$ u find out by considering that $f$ should become continuous