Understanding the delta epsilon definition of limit is major milestone in the initial stage of development of non-trivial mathematical maturity. However, one who knows topology is well aware that limit and convergence can be defined in much more generality in the context of topological spaces. In the initial stage, limits were studied almost exclusively on the reals, as limits of a discrete sequence or of a real function wherein the input variable tends continuously to some value. The limit’s being equal to infinity was also separately defined.
There is also that sequences were not generalized to nets in the 1920s and ultrafilters in the 1930s after point-set topology had reached a certain stage of maturity. They are of course more abstract and in some sense seemingly elusive, but with a certain level of depth of thought, one should view them as more or less a natural development or generalization. Another major generalization was going from a concrete metrizable space to an abstract topological space for which convergence of a sequence or net is defined in terms of neighborhoods.
It is not uncommon for a student to initially struggle with topology, getting lost in the abstract formalism devoid of any meaningful intuition for the definitions. I certainly was one of them. So here, I will explain how one would naturally arrive at the definition of open and closed sets, neighborhoods, topological spaces, and convergence.
Intuitively, a neighborhood of a point is a set containing it and its most immediate surroundings. An open set is a set that is a neighborhood of all of its points. A sequence of points in a topological space converges to a point in it if and only if for every neighborhood of , there exists an such that for all , lies in that neighborhood. This is basically saying that no matter how small a set containing , as long as it contains it and its most immediate surroundings, the sequence past some point will be constrained to that set.
What are the requirements for a neighborhood of a point then, without reference to the definition of open set? Well, of course, it must contain the point itself. If it includes a neighborhood of , it should also be a neighborhood, which is also a no-brainer. The last two requirements are more non-trivial. They are that
- The intersection of two neighborhoods of is also a neighborhood. This makes sense because intuitively the neighborhood of a point contains the most immediate vicinity of it. The intersection of two sets which satisfy this containment criteria naturally also satisfies it.
- If is a neighborhood of , then there exists a neighborhood of , such that is a neighborhood of all the points of . Crudely speaking, this is saying that any set containing the most immediate vicinity of an arbitrary point should be regarded as containing the most immediate vicinity of every point in the most immediate vicinity of .
The usage of “most immediate vicinity” is imperfect or sloppy in the sense that in topological spaces generally thought of, such as , there is no strictly most immediate vicinity. Via the ball of radius for large, one gets a very immediate vicinity, but no matter how large is here, there always exists an even more immediate vicinity. Yet the intersection of all such balls gives the singleton set itself, which does not contain any immediate vicinity of and is thus not a neighborhood.
Naively, one might expect the intersection of infinitely many neighborhood to also be a neighborhood, especially when one thinks in terms of most immediate vicinity. Thus, we modify “most immediate vicinity” to “all arbitrarily immediate vicinities”. Then, it occurs that if we take a sequence of successively strictly more immediate vicinities such that for any set that counts as an immediate vicinity, there is a strictly even more immediate vicinity in the sequence, their intersection should not necessarily contain all arbitrarily immediate vicinities. One notes that mathematical induction, when applied to intersection of two finite sets, extends to intersection of a finite number of sets but not to infinity, and there is analogous is measure theory, with countable additivity, usually involving taking limits of sequences of non-negative numbers, much harder to prove than finite additivity.
As for definition of topological spaces via open sets, one who has studied topology should remember that a topological space on , as a family of subsets of , must contain both . The universal set of the space should be regarded as a neighborhood of all the points in it. Another way of viewing this is that if were not in the topological space, then some point in the space would not have a neighborhood, which would be absurd. As for , because there are no elements in it, it need not be a neighborhood of any point, and thus, nothing here violates the definition of an open set. This is like how you initialize your boolean variable to true when evaluating logical conjunction on a list of booleans. If there is no boolean to be evaluated, you should return true.
As for closure under unions, including uncountably many, this is obvious since for each point in the union, the union must contain one of the open sets passed as an argument to the union operator, which by definition of open set, is a neighborhood of it. For finite intersection, we simply have that the intersection is a neighborhood of all the points in the intersection.
After defining topological space and convergence in it, one can define the set of accumulation points of a subset of the set on which the topological space is defined to be the set of points sequences in can converge to. The set of accumulation points of is sometimes also called the closure of or . is a closed set iff .
Another fact of convergence in point-set topology that might be counterintuitive to a person who has learned convergence of sequences in the reals, where a sequence converges to either one point or none, too “dogmatically” is that in a topological space, one can have convergence to multiple points. Even on , there is the trivial topology . Every point in this topological space has only one neighborhood or only one immediate vicinity, which is all of dimensional space, wherein one can go arbitrarily far away if one thinks in terms of the Euclidean metric. Here, in the coarsest topology possible, every sequence converges to every element of dimensional real space.
Two points are called topologically indistinguishable if a set is a neighborhood of iff it is a neighborhood of . One can easily verify that topologically indistinguishability is an equivalence relation. Moreover, if two points are topologically indistinguishable, a sequence converging to one of them necessarily also converges to the other. In the trivial topology, all points are mutually topologically indistinguishable.