I am reading Munkres Topology Chapter 13, in which some examples of bases of topologies are given. One of the examples compares the two possible bases (a, b) and [a, b) on the real line.

I understand that both (a,b) and [a,b) satisfy the definition of basis (intersection between two element has the same form and all elements together form the whole real line). 

However, in the definition of a topology (as a subset of the power set of the real line), every element is called an open set. Clearly this "open" definition is different from the traditional "open" definition as [a, b) cannot be open in the traditional definition.

Furthermore, I can see that even [a, b] can be "open" in the real line (in the topology generated by [a, b]) because any non empty intersection also has the form [a, b] and the whole union makes up the whole real line. But this makes no sense using the traditional definition where [a, b] is obviously closed.

My question is why every element of a topology is named "open"? What is the intuition behind this name?