How is an open set defined without referring to any distance function in a topology?

Question

I am currently studying general topology. The definition given by Royden looks very confusing to me. It says that the elements of a topology is called open sets without actually defining what exactly an open set is. If one can simply names things, can I call elements of a topology closed sets or maybe even elephants, please? My guessing here is that the concept of an open set in topology is rather flexible in the sense that one can simply call certain subsets of a set open sets so long as the topology satisfies the closure properties. Could anyone help me understand how exactly an open set is defined in the context of topology, please? Thank you!

Answer 1

That is exactly what it looks like, the elements of a topology are calles open sets.

You must understand that this is a definition of a topological space. The definition is composed of a set of axioms, and any structure you may have that satisfies the axioms may be called a topological set. For example, the family of all open intervals of $\mathbb R$ is a topology on $\mathbb R$. In this topology, a set is open (topologicaly) if and only if it is open (in the metric-defined way of being open). This is also the reason why the topology sets are called open (the name fits with the way it is used elsewhere).

You must understand that every time you define a new mathematical structure, you are, at first, simply playing with words. The definition is abstract and only says "if something follows rules 1, 2, 3 and 4, then I call it $x$." The definition does not know whether it is useful or even if anything satisfies it, it is just there. Of course, most things we define are useful and exist (in some way), just like when we define topological spaces. In the definition, we try to model some aspects of how open sets on metric spaces behave.