Trying to Understand the First Paragraph of “A Concise Course in Algebraic Topology”

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I. 

Wikipedia is here, and quite helpful in expanding on this definition. Here is my attempt to rewrite it in my own words, fleshing it out in ways that are helpful and natural to me.

What is a topological space?

Imagine that you have some collection of points that are in the Cartesian plane. Maybe it’s a circle that’s centered around the origin. Maybe it’s a line, like y = 2x + 3. Maybe it’s a grid of points falling perfectly along the chassis of the plane.

In a way, we can also imagine the city that you live in or the state or country as a collection of points. My apartment is one of these points in space, and I’m part of a neighborhood — Washington Heights. But there are other ways of talking about where I am. I’m not just in Washington Heights, but in NYC, in NYS, in USA, on Earth. There are points, arguably, south of where I live that are a part of two neighborhoods — Washington Heights and Harlem. These neighborhoods aren’t coextensive, but they do have some overlap.

Maybe there are some folks living out in the sticks that are part of no neighborhood. They’re off the grid, so to speak.

Let’s imagine a computer program that knows how to assign every point in a map of NYC to the neighborhoods to which it belongs. Let’s call that program N. You give N a point, and it gives you a list of neighborhoods.

Imagine if we let N loose on any of those spaces that we were talking about before — the disk, the grid of integer points, the portion of a line — and then let’s say that any of those spaces is a topological space if when N is applied to the space, nothing weird happens. Here is a checklist of four weird things that could happen:

  • You apply the computer program N, but there’s a place that gets assigned neighborhoods that it’s not in. Like, maybe the program assigns a spot in the middle of the Hudson River to Washington Heights, even though a random spot in the river isn’t really a part of our neighborhood.
  • Imagine that N says that my apartment is in Washington Heights, but not in NYC. That’s messed up. Bad N. If you can imagine a place like that, it’s not a topological space.
  • If I lived a bit further north, I might be a part of two neighborhoods: Washington Heights and Inwood. N should assign me not just to Washington Heights and to Inwood, but to a new, hybrid neighborhood: Inwoodton Heights. If N can’t really invent such a neighborhood, then I don’t really live in a topological space. This one starts moving away from my regular intuitions about neighborhoods, clearly…
  • …and so does this last test. As I keep on mentioning, my apartment is in a neighborhood Washington Heights. If N assigns my apartment to WaHi, then it should also report that there’s another neighborhood that I’m a part of that’s more local than WaHi, like my block. My block has the property that (a) I’m in it, it’s my neighborhood and also (b) everywhere on my block is also part of Washington Heights. If there isn’t such a place, I don’t live in a topological space.

Moving away from apartments and cities, let’s think about the x-axis. Real estate agents who are trying to get people with $ to move to a place decide what neighborhoods mean in the real world. But what does a neighborhood mean on the x-axis? Let’s define a ‘hood of a point (like 3.1) to be any open interval (like between 3 and 5) that includes the point.

Let’s make sure that our computer program N would work alright in such a place:

  • If you’re a neighborhood of a point, that point is in it. Yay!
  • If you’re in an interval (like between 3 and 5) and that interval is in a bigger interval (like between 1 and 6) then you’re also in the bigger neighborhood.
  • If you’re part of two neighborhoods (like between 3 and 5 and also between 3.05 and 7) then you’re also part of the overlapping neighborhood (between 3.05 and 5).
  • If you’re part of a neighborhood (like between 3 and 5) there is a smaller, more local neighborhood that you’re part of (like between 3.05 and 4.95) and, more to the point, there always will be.

Huzzah — it’s a topological space!
II.

One more twist: this way of thinking about topological spaces is not standard any longer. People prefer to talk not about neighborhoods but instead about open sets. So we have to make sense of that, even though it’s been nearly a decade since we took Real Analysis.

Another wise wikipedia page says, “an open set is an abstract concept generalizing the idea of an open interval in the real line,” which makes sense. It tells you the delimit of a set, but it doesn’t include its boundary.

This isn’t enough, though. Since we defined a topological space in terms of neighborhoods, we want a definition of “open set” that relates “open set” to neighborhood. We need to connect these two concepts.

[Here’s where I got a little bit lost, so I went looking for help from another source. Google google, ended up at the Math Stackexchange.]

It doesn’t take much to merge the two concepts, as far as I can tell from what I’m reading. To fit the rules of neighborhood assignment, your potential neighborhood has to pass the following test: if a point is assigned a particular ‘hood, the point actually has to be in that ‘hood. (Fancy talk: if N is in N(x), then x has to be in N.)

An open set adds one slight additional requirement: if you’re in N, then N’s your neighborhood. All this eliminates is the possibility that a neighborhood is “too big,” including not just Washington Heights but also a random stretch of the Hudson River where nobody lives.

This is a definition that will do the trick for us, and allow us to connect the old-fashioned (but apparently useful for beginners like myself) definition of topological space to the new and trickier (but apparently useful for topology pros) definition that is couched in terms of open sets.

Here’s the open set version: still imagine a map of some space (like NYC) and still imagine that the map contains a bunch of points (like my apartment and other peoples’ apartments) and still imagine that there are ways of grouping those points that come with the map (like neighborhoods)…

…or don’t, and instead imagine some section of the x-axis and the Cartesian plane, and imagine that there are ways of making subsets of that section of the line or plane…

And now there are three rules about the neighborhoods (or subsets):

  • The empty set is in it — i.e. the neighborhood of nobody is a neighborhood
  • You can’t escape the space via the union of subsets — i.e. mush together any of the neighborhoods (subsets) and you’re still in a neighborhood (subset)
  • You can’t escape the space via the finite intersection of subsets — i.e. be in as many of the neighborhoods (subsets) as you wish to at once and you’re still in a neighborhood (subset)

Not entirely obvious to me yet why these two definitions of topological space are equivalent. I see that they both have the intersection, I see how the intersection of disjoint sets would imply that the empty set is in the topology (but what’s wrong if there’s only one set in the topology according to the neighborhood definition?), and I guess the superset/subset axioms must cover the “closed under union” axiom of the open set definition? I’m going to let that slide for now.

III.

What are some examples and non-examples of a topological space? We already mentioned the x-axis, equipped with open intervals.

In fact, for things like the x-axis — metric spaces — the textbook page already tells us that we can think of topological spaces as things where the neighborhoods are like open intervals. They’re open disks, or open spheres, that act precisely in the way you’d expect them to. No funny business, no weird stuff.

As the textbook also says, we should think of this all as an attempt to point at what it takes to have a space that captures our feelings about “nearness.” So if something’s near, there’s always something nearer, that seems to be the most important part to me.

But whenever there’s a new idea, we need some examples and contrasting non-examples to set our heads on straight, and this is no exception.

The x-axis — i.e. the real line — is an infinite collection of points, but there’s no reason why a topological space needs to be infinite. And if our map of points is finite, there’s no reason why our computer program — our function — N has to be anything but a list of neighborhoods.

So, suppose you have four points in your space: 1, 2, 3 . What neighborhood assignments would result in this being a topological space, and which wouldn’t? Here is a helpful image from wikipedia:

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  • If you care your space into two sets — {} and {1, 2, 3} — then you haven’t really done much carving, but that’s a topological space, the trivial topology. That sort of captures the idea of nearness in an absolute way — everything is near, nothing is not, like a terrible party. (No idea if that analogy makes mathematical sense, by the way.)

Thinking about the function N and the first set of axioms: the same and only neighborhood has been assigned to the three points and they’re part of it, there are no supersets or subsets, so trivially we’re done.

  • In contrast, if you carve your space into {}, {1, 2, 3}, {2} and {3} (bottom left) that’s not a topological space. From the point of view of the second definition, the issue is that the union of {2} and {3} isn’t included, so it’s not a topology. From the point of view of the first, the problem is that {2, 3} is a superset of {2}, and every subset containing a neighborhood must itself be a neighborhood. (Is this right?? It doesn’t feel right.)
  • What if your topology contains {}, {1}, {2}, {1, 2}, {1, 2, 3} (middle left)? The second definition is indeed sort of easier to use for these discrete examples. It’s pretty clear: the union of {1}, {2} is {1, 2} and that’s in.

I need more examples, and wikipedia has a few more:

  • Let the space be all the integers, and let the collection of subsets be all the finite subsets of the integers, any list of them, plus the set of all the integers. So, for example, if you have a given point like -5, there would be all these open intervals that we’d want to say are the neighborhoods of -5, like “integers between -10 and 2” and “integers between 0 and -10000,” and we’d also say “all the integers.” That wouldn’t work, though, because you the union of a bunch of these subsets isn’t necessarily part of the topology. Take the union of all the finite sets that don’t contain zero — {-2, -1, 1, 2}, {1, 5, 6}, etc. — that’s an infinite set, but there’s only one infinite set in the collection (all the integers) and this can’t be that infinite set. Therefore, it can’t be in the collection, and this can’t be a topology.

The tricky thing for me is relating this all to the talk that a topological space preserves our notion of “nearness.” Is it possible to relate each of these axiom failures to a failure of our notion of nearness? Personally speaking, my notion of nearness no longer operates when our space has just four points. Is there a good way to think about this that I’m missing?

TO BE CONTINUED, WHEN I TRY TO READ THE SECOND AND THIRD PARAGRAPHS OF THIS TEXT AS I ATTEMPT TO UNDERSTAND A PROOF OF BROUWER’S FIXED POINT THEOREM. STAY TUNED

 

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