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?
Best Answer
I believe that this terminology predates topological formalism. When nineteenth century mathematicians began laying the foundations of what we call today analysis and topology, they noticed that what we call today open sets in the standard topology on $\mathbb R$ have nice properties (closed under unions, finite intersections, etc...).
When considering other spaces besides $\mathbb R$, it is useful to determine the least amount of structure to apply to that space to specify its unique properties. Consider, for instance the circle. It has properties that are different from any subset of the real numbers. How much information do we need to know about a space before we can declare that it has the properties of a circle? Certainly this depends on which properties in which you are interested, but early topologist discovered that not only can many interesting properties can be determined by simply specifying which subsets of that space have the same behavior as open sets in $\mathbb R$, but also that proving that these properties hold is not terribly difficult in most cases. So by specifying a relatively small amount of information about the space, we can describe it in great detail with relative ease.
I believe that the name "open set" simply carries over from the standard topology on the reals where the description of open relates more clearly with the English definition of the word. Nevertheless, as I often have to explain to my non-mathematician friends, if an English word has a mathematical definition associated with it, that definition need not have any relation to the English definition of the word (though it usually does to some extent). So even though we call sets open that can't be described as open under even the most abstract of English definitions, it is the word we use in math, and it's here to stay.