The definition of a topological space might be one of the definitions with the largest “level of abstraction” to “mathematical maturity when first encountered” ratio. The usual definition (as in Munkres) is:
A topology on a set \(X\) is a collection \(T\) of subsets of \(X\) having the following properties:
- \(\varnothing\) and \(X\) are in \(T\)
- The union of the elements of any finite subcollection of \(T\) is in \(T\)
- The intersection of the elements of any finite subcollection of \(T\) is in \(T\)
A set \(X\) for which a topology \(T\) has been specified is called a topological space. The elements of \(T\) are then said to be open sets.
Many students scratch their heads when seeing this for the first time and ask for an explanation, perhaps there is some intuition that might shine a light on why such a definition for an open set is useful and interesting. The usual answer one hears is to think about the open sets of \(\mathbb{R}\), or more generally, the open balls of a metric space that one has encountered before and notice how this definition of an open set generalizes those examples and ends up being useful because of its applicability in many instances. This is certainly enough, at the end of the day one might have already worked with groups and seen how in that case studying a generalized notion of symmetry leads to many interesting results and applications. But could there be some interpretation? Just as in the case of groups we quickly learn Cayley’s theorem and find out that every group is a group of symmetries, is there also such an interpretation for what open sets represent?
There turns out be just such an interpretation, and it comes from a computational point of view (in fact there’s another geometric interpretation1 which might be even more illuminating but I want to talk about the computational one!). There is a popular MathOverflow thread2 where this is discussed and digging deeper leads one to a handbook article by Michael Smyth3 which was one of the first researchers to develop this idea. I wish to present this knowledge in plain language to help future confused students and cement the concepts in my own mind.
Let’s start with the set \(X\) and a subset of it \(U\). Now imagine for simplicity’s sake that we have a program that takes an element \(x\) of \(X\) and halts when \(x \in U\), but runs forever otherwise (this can be formalized). We then say that \(U\) is an observable property and corresponds to the program mentioned above.
It follows that if we have a collection of observable properties \( \{ U_i \}_{i \in I} \), then \( \cup U_i \) is also an observable property, since if \( x \in \cup U_i \) it follows that \( x \in U_i \) for some \( i \) and we can get its corresponding program that halts when given \(x\) as input. So the overall program can be generated by running the programs for each \( U_i \) in parallel and making it halt as soon as one of the “subprograms” halts.
On the other hand, if \( U_1 \) and \( U_2 \) are observable properties, then so is \( U_1 \cap U_2 \) by using the same logic as above but now waiting for both the programs to halt, not just one. Note how finiteness is needed here, to see this imagine a collection \( U_1, U_2, \dots \) of observable properties with programs that halt in \(1, 2, \dots \) seconds respectively. If we wanted to see whether an element is in the infinite intersection by running all programs in parallel we would never finish waiting!
These are precisely the properties that we use to define open sets! Therefore, when we put a topology on a set, it’s as if we are choosing a lens and defining what we are able to “see” which also has implications for the “shape” of the set in the geometric viewpoint. For instance, in the standard topology of \( \mathbb{R} \) we pick the open intervals as a basis, so it’s as if we strip all information out except for the ordering and can only see whether an element is between other two other elements and nothing else. So in this point of view a set such as \([a, b)\) is not open because the fact that \(a \in [a, b) \) is not “observable” (imagine a program drawing vertical lines at \(x=a\) and \(x=b\), giving it the point a and asking it whether it is between the two lines, only for it to keep zooming in forever without being able to see whether this point is between both the lines).
This also answers the famous question of why not define a topology in terms of the closed sets, which can be done by just swapping infinite unions for finite unions and finite intersections for infinite intersections. In this point of view, closed sets correspond to sets of elements that don’t have a particular observable property (which makes clear the dual nature of open and closed sets and shows that it is not really interesting to define a topology in terms of closed sets), and what would happen if we took the closed intervals \([a, b]\) of \( \mathbb{R} \) and tried to use them as a basis for a topology? We see that this would imply that we can now “see” individual elements (take the intersection of two touching closed intervals) so this would just give us the discrete topology, the finest of the topologies of a set that makes every function continuous, so again, we’re in a less interesting world.
Other topological concepts turn out to also have computational interpretations, to which I refer you to the references. If you find cool applications of this point of view let me know!