In class last week, we learned about the uniqueness of limits. Specifically, the limit of a sequence, when it exists, must be unique. But what if I told you that there existed a place in mathematics where this is not always the case? Today, I’d like to briefly define that context and then illustrate some basic examples. Finally, I will note how our new and old mathematical analysis knowledge converge (forgive the pun) when we look at sequences from the perspective of this amazing field of mathematics called a topology.

 

A topological space consists of a set and a topology , which is a collection of subsets of called . For to be a topology, it must satisfy three requirements:

1. 
2. Given any collection of sets in , must contain their union.
3. Given any finite collection of sets in , must contain their intersection.

Let’s begin by looking at the trivial topology on where Does this satisfy the requirements of a topology? Let’s check:

1. Trivial
2. contains the unions of all of the collection of sets in :
   • 
   • 
   • 
3. contains the intersection of all of the finite collections of sets in :
   • 
   • 
   • 

Thus we may conclude that the trivial topology is, truly, the trivial topology.

Here are some visual examples of and elements in are circled:

Source: https://en.wikipedia.org/wiki/Topological_space

Everything circled is in . So the upper left diagram is the trivial topology on X previously described. Notice that the bottom two diagrams are not topological spaces. Can you see why? For the bottom left image, does not satisfy requirement 2 because it does not contain the element {2,3}. In the bottom right image, does not satisfy requirement 3 because it does not contain the element {2}.

Now that you are more familiar with the notion of a topological space, what about something more interesting? Let’s look at the topology :

1. Trivial
2. The union of all subcollections of sets in will either be the reals, the empty set, or {1}.
3. The intersection of any combination of these three elements will either be the empty set or the reals or {1}.

In fact, we find that any set composed of is a topology on .

 

So now, where does this leave us? Perhaps a bit overwhelmed. I know I certainly was the first time I ran through the material. But maybe the mind-bending result of sequence behavior in topological spaces will keep you hooked on topologies forever. At the beginning of this post, I reminded you that, as far as our mathematical knowledge takes us, limits are unique. But this is only the case for the standard topology on which consists of arbitrary unions and finite intersections of open intervals.

Let’s look at the sequence (1/n)=(1,(1/2),(1/3),(1/4),…) We know that in with the standard topology and our usual -neighborhood definition of convergence, this sequence only converges to 0. But what about in a different topological space? Well, in a topological space, . Let’s consider the topology . Here, if , then , or for some B. Take T = {} which is in . will never eventually be in {} because neither or 0 is in the sequence. This means that does not converge to 0 in this topological space. We have now found a sequence in that converges in some topologies but not in others.

Are you ready to have your head blown even more? Let’s look at once more, but this time using the trivial topology. certainly converges to 0 in the trivial topology, but it also converges to all of the real numbers. Given any , the only with is , and the sequence is eventually in $\mathbf{R}$, so for all converges to a!

While we don’t know much about them, topologies are a fascinating area of mathematics, and while I have only scratched the surface in my post today, maybe I have done enough in this post to pique your curiosity for more.