Can we call every set a topological space?

Question

I am new student in the course of topology so i am confused whether every set is a topological space ? If anyone could help me out for this I will remain thankful.

Answer 1

As coffeemath states, a topological space is a set equipped with a collection of subsets of that set, called the open subsets, that satisfy certain laws. So a set by itself is not a topological space. That said, every set gives rise to a topological space. In particular, it gives rise to the discrete topological space where we take the set of all subsets, i.e. the powerset, as the set of open subsets. This is the most natural way to view an arbitrary set as a topological space. There are other topological structures that you could induce given an arbitrary set. For example, the indiscrete topology which takes only the empty set and the set itself as open subsets. However, in some sense, this isn't too different from collapsing the set to a single point (assuming it is non-empty).

It's extremely common in mathematics in general to talk about a notion of "structured set" while leaving the "structure" on the set implicit in examples when it is deemed "obvious" or "implied". So, for example, you'll often see $\mathbb{R}$, the set of real numbers, being called a topological space by which is meant $\mathbb{R}$ equipped with the set of arbitrary unions of open intervals as the open sets (i.e. the topology induced by the metric structure on $\mathbb{R}$). There are many other possibilities (such as the discrete or indiscrete topologies, but many others besides those).