“That structure was just guess work, but we seem to have guessed well.”

This interview with Alan Baddeley, one of the researchers responsible for the contemporary notion of “working memory,” is fantastic. I’m not up to speed on all the psych concepts yet, but there are a lot of juicy bits.

First, you can’t get into research like this any longer:

When I graduated I went to the States for a year hoping, when I returned, to do research on partial reinforcement in rats. But when I came back the whole behaviourist enterprise was largely in ruins. The big controversy between Hull and Tolman had apparently been abandoned as a draw and everybody moved on to do something else. On return, I didn’t have a PhD place, and the only job I could get was as a hospital porter and later as a secondary modern school teacher – with no training whatsoever! Then a job cropped up at the Medical Research Council Applied Psychology Unit in Cambridge. They had a project funded by the Post Office on the design of postal codes and so I started doing research on memory.

On his “guess work”:

I think what we did was to move away from the idea of a limited short-term memory that was largely verbal to something that was much broader, and that was essentially concerned with helping us cope with cognitive problems. So we moved from a simple verbal store to a three component store that was run by an attentional executive and that was assisted by a visual spatial storage system and a verbal storage system. That structure was just guess work, but we seem to have guessed well because the three components are still in there 30 odd years later – although now with a fourth component, the ‘episodic buffer’.

What if he had guessed something different, and if that different guess had held up decently? Psychology is at a local maximum, but it seems to me that there’s very reason to think that our current conception of the mind is at anywhere near a global maximum, the most natural and useful way of conceiving of things.

On that:

The basic model is not too hard to understand, but potentially it’s expandable. I think that’s why it’s survived.

Here is the context for their description of working memory, as distinctive from short-term memory:

I suppose the model came reasonably quickly. Graham and I got a three-year grant to look at the relationship between long- and short-term memory just at a time when people were abandoning the study of short-term memory because the concept was running into problems. One of the problems was that patients who seemed to have a very impaired short-term memory, with a digit span of only one or two, nevertheless could have preserved long-term memory. The problem is that short-term memory was assumed to be a crucial stage in feeding long-term memory, so such patients ought to have been amnesic as well. They were not. Similarly, if short-term memory acted as a working memory, the patients ought to be virtually demented because of problems with complex cognition. In fact they were fine. One of them worked as a secretary, another a taxi driver and one of them ran a shop. They had very specific deficits that were inconsistent with the old idea that short-term memory simply feeds long-term memory. So what we decided to do was to split short-term memory into various components, proposing a verbal component, a visual spatial one, and clearly it needed some sort of attentional controller. We reckoned these three were the minimum needed.

In other words, short-term memory was assumed to be unitary. Baddeley and Hitch figured that it could have three independent components. Since short-term memory was running into trouble, it sounds like they kind of rebranded it as working memory.

Are there other differences between short-term memory and working memory besides its multi-component nature? I think so, but I’m currently fuzzy on this. Another thing that I’ve read suggests that short-term memory was seen as just on the pathway to long-term memory, so it was essentially part of a memory-forming pathway. Working memory is supposed to play a broader range of roles…I’m confused on this, honestly.

Finally, here’s a solid interaction:

Your model with Graham Hitch has a central executive controlling ‘slave’ systems. People sometimes have a problem with the term ‘slave’?
This is presumably because people don’t like the idea of slavery.

 

First instance of the term “Working Memory”

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“When we have decided to execute some particular Plan, it is probably put into some special state or place where it can be remembered while it is being executed. Particularly if it is a transient, temporary kind of Plan that will be used today and never again, we need some special place to store it. The special place may be on a sheet of paper. Or (who knows?) it may be somewhere in the frontal love of the brain. Without committing ourselves to any specific machinery, therefore, we should like to speak of the memory we use for the execution of our Plans as a kind of quick-access, “working memory.”

From Plans and the Structure of Behavior (1960)

Some of the things I stand for in education, these days

When we critique an idea, we should critique the best version of it.

When we critique a pedagogical idea, we should critique the idea itself, not its misinterpretations. (Unless we’re saying the idea is easy to misinterpret.)

Most of the time when a pedagogical idea is critiqued, it’s critiqued for bad choices the teacher might make that have nothing to do with the idea itself. When we imagine an idea we don’t like, we imagine a classroom that we don’t like. That’s not fair, though.

Every pedagogical idea gets misinterpreted.

Nobody knows how to make ideas about good teaching scale, so we might as well talk about what good teaching actually looks like, without worrying about how the truth will or will not be misinterpreted at large.

You and I might teach in very similar ways but have wildly different ways to describe how we teach. The fiercest debates in education are also the vaguest. When you get down to classroom details, or even the tiniest bit of additional specificity, a lot of disagreement vanishes. It’s not that these debates don’t matter, it’s that they are highly theoretical.

OK maybe these debate don’t really matter.

Teachers have access to classroom details and specificity. One way that teachers can contribute to the knowledge base of education is to resist the urge to move to generalization, to spend some more time in the greater specificity that classroom life encourages. This is what teachers can uniquely contribute.

Non-teachers will often tend to think about scale. Teachers usually don’t, and this is something else that we can contribute.

Some people will tell you that being a jerk is important, as long as you’re being a jerk to the right people. Those people are jerks. Stupid jerks.

I think the core of what I do is trying to have a detailed understanding of how my students think about the stuff I’m teaching, and also of how they could think about the stuff I’m teaching. And then I’m trying to create as many opportunities for them to think about that stuff in the new way.

I care about my students’ feelings, a lot.

The way I see it, teaching is best understood through its dilemmas and tensions. This is not a new idea, but it’s one that hasn’t been sufficiently explored. And I’m very suspicious of people who claim to have resolved one of these tensions or dilemmas.

We shouldn’t ad hoc create rules for political discourse that only apply to people who we disagree with.

Debate is essentially performative. I mean, it doesn’t need to be, but it usually is.

Trying to understand someone and how they think is an almost complete improvement over debate in every way.

Lots of research is interesting. It’s especially fun to try to figure out how two different perspectives on teaching can fit together.

Teaching is a job, but it can be a great job if you like ideas and little humans.

Writing and reading should be fun and interesting. When writing instead aims to be useful, it’s usually not useful either. And 99% of writing about teaching is supposed to be useful.

At a certain point you have to decide if you’re trying, primarily, to change the world or to understand and describe it. It’s great that some people are trying to change the world, but I think inevitably those people end up having to be less-than-honest at times, for the sake of their projects or reputation. Personally, I prefer to understand and describe it when I write or think. (I’m not opposed to helping people, though!)

Blogging is dead, but it’s all we got for now, so let’s keep at it.

I like to think of my teaching life as a bubble, and I like to think of all other aspects of my life as little bubbles too. And a question I ask myself frequently is, can I grow these bubbles? Wouldn’t it be something if all the other bubbles could float and sort of merge into each other, turn into just one big bubble that encompasses everything? I feel like that would be nice, someday.

A New Study about Gender and Pay Gaps

I learned about this via Marginal Revolutions and Freakonomics. Briefly, Uber keeps an enormous amount of data on its drivers, allowing economists to study the different ways that women and men are paid. The Freakonomics folks interviewed an Uber economist:

LIST: So we have mounds and mounds of data. We have millions of drivers. We have millions of observations, and 25 million driver-weeks across 196 cities. So just the depth of the data and the understanding of both the compensation function and the production function of drivers gives us a chance to — once we observe if there is indeed a gap — gives us a chance to unpack what are the features that can explain that gap.

They did find a pay gap that broke down by gender — 7%:

LIST: We found something very surprising. What you find is that men make about 7 percent more per hour on average …

DIAMOND: … which is pretty substantial.

LIST: For doing the exact same job in a setting where work assignments are made by a gender-blind algorithm and pay structure’s tied directly to output and not negotiated.

Was it because of discrimination on an individual level? They don’t think so:

DUBNER: Right. So let me just make sure I’m clear. You’re saying there’s no discrimination on the Uber side, on the supply side, because the algorithm is gender-blind and the price is the price. And you’re saying there’s no discrimination on the passenger side. So does that mean that discrimination accounts for zero percent of whatever pay gap you find or don’t find between male and female Uber drivers?

LIST: That’s correct.

The interview is long and full of juicy details and tough questions. I haven’t read the whole thing carefully yet, but it’s fascinating all the way through:

DUBNER: What is the overall driver attrition rate? I don’t know whether it’s measured in six months or a year, or whatever.

DIAMOND: Yes, six months is what we’ve been looking at.

LIST: More than 60 percent of those who start driving are no longer active on the platform six months later.

DIAMOND: So the six-month attrition rate for the whole U.S. for men is about 63 percent, and for women it’s about 76 percent.

DUBNER: Wow. So that would connote to me, an amateur at least, that maybe this gender pay gap among Uber drivers is reflected in the fact that women leave it so much more. Maybe it’s just a job that on average, women really don’t like. Is that measurable?

There’s a whole discussion about that, and a lot  of other things too besides.

So what does explain the 7% pay gap, in the end? They have theories, foremost among them is that men drive faster than women:

LIST: That’s right. So after we account for experience now we’re left scratching our heads. So, we’re thinking, “Well, we’ve tried discrimination. We’ve done where, when. We’ve done experience. What possibly could it be?” What we notice in the data is that men are actually completing more trips per hour than women. So this is sort of a eureka moment.

DUBNER: They’re driving faster, aren’t they?

HALL: Yeah. So the third factor, which explains the remaining 50 percent of the gap, is speed.

It’s not hard to speculate about how something like how quickly men vs. women tend to drive to possible sources of systemic cultural discrimination. (Are men more confident drivers? Are they less fearful of the law? etc.)

Still, what these economists are finding is that (a) the pay gap is persistent, even in the face of an equalizing pay structure and (b) possible factors explaining the gap will not be simple to address. For example, one source of the gap could be that men tend to work more hours, gaining more experience which pays dividends later, so that one source of the Uber gap is that men are getting paid for experience, not something that you can easily address:

DIAMOND: I think this is showing that the gender pay gap is not likely to go away completely anytime soon. Unless somehow, things in our broader society really change, about how men and women are making choices about their broader lives, than just the labor market. But it’s not also a worry that the labor market is not functioning correctly. It makes sense to compensate people who are doing more productive work. It makes sense to pay people more if they work more hours. I mean, I don’t think those are things that we would ever consider thinking should be changed because that they’re a problem. Those are just real reasons that productivity can differ between men and women. And we should compensate people based on productivity.

What would the implications of this finding be? It’s not that individual discrimination isn’t responsible for the pay gap in general — this would likely depend on the field and the job, right? — but that there are deeper factors at play that might explain a pay gap between men and women.

This shouldn’t really surprise anyone working in education. Men and women teachers are paid according to the same standardized salary schedule in public schools. If you pooled all the male and female teachers, though, you’d see that there is a gender pay gap because of disproportionate numbers of male/female teachers in elementary vs. middle vs. high school. Men make more not because administrators choose to pay them more — largely, it’s because men choose to teach older kids more than women.

There’s no easy way to address this discrimination, if it’s even quite right to call this discrimination. Certainly it’s possible (likely? I don’t really know) that cultural factors partly explain the choices of men and women. At a certain level, though, this is irrelevant. Do we want to mess with the choices of men and women about where and how to work?

(This quickly gets tangled in questions of diversity and representation. Is it a policy priority to ensure that men and women are represented in the teaching force at numbers that are proportional to the students they instruct?)

There might be structural pay gaps that outstrip what can be explained by discrimination. Whether biological or culturally constructed, there might be persistent and not-bad differences between men and women. As I continue to think about this, it seems to make sense to me that we’d want to continue to battle discrimination while also thinking of ways to address inequities using other angles, besides battling pay gaps. Past a certain point, of course.

One last juicy morsel, at the very end of the interview. As Uber introduces tipping it seems that drivers make LESS and that the pay gap narrows somewhat because women are tipped more:

LIST: Yeah, I think when you look at the tipping data in general, you do find a tilt in favor of women compared to men in general. We’ll have a tipping paper for you in a few months. Because the economics of tipping is sort of wide open, and we’ll have a paper just like this one called something like “A Nationwide Experiment on Tipping.”

DUBNER: Right.

LIST: And we’ll do the tipping roll out and show you how earnings change with the introduction of tipping. And the earnings actually go down a little bit. They don’t go up after you introduce tipping.

DUBNER: Now how can that be?

LIST: What happens is the supply curve shifts out enough to compensate the higher tips. And when the supply curve shifts out, the organic wages go down. And what you have is drivers are underutilized. So what I mean by that is typically they’ll sit in their car empty 35 percent of the time. With tipping, maybe it’ll go up to 38 percent of the time.

DUBNER: In other words, the wage declines because more drivers think they’re going to make more money since tips are now included, but that increases the supply of drivers, which means there’s less demand to go around.

LIST: Exactly. That’s perfect.

 

What I wish the NCTM journals were

It’s hard for me to separate what I like doing as a writer from what I enjoy reading. Back when I was critiquing the NCTM journals it felt like I was being sort of vague about what I wanted…well, I had to be vague, because what I want to read are roughly the kinds of things that I write. (Emphasis on kinds; I don’t want to read me, I want to read you.)

But just to put the pieces together, here is what I would love to read in an NCTM (or any other) math education journal:

Toss in a few features — crosswords, math problems, a curriculum resource, etc. — and this is the sort of thing that I, Michael Pershan, would be interested in reading. I have no idea who else would read or pay money for this sort of thing, so please NCTM don’t rashly decide to change the journals to more closely resemble my blogging output. But this is close to what I imagine would be my ideal publication, i.e. the sort of pretentious garbage that’s my calling card.

Reading “A Root Cause of the Teacher-Diversity Problem”

Check it out here, it’s great.

So far the main effect of having a second kid in the house has been to make me, in pretty much every way, a little bit dumber. You know what I mean by dumber: not as clever, etc.

Anyway, one of the ways in which I’m dumber is that I’m a worse reader with worse impulse control. So last night I found myself reading an article right before bed (another mistake) and having trouble understanding a perfectly clear article by Melinda D. Anderson (the one I mentioned). And then I had the questionable decision to tweet the author my questions.

I really feel for journalists on Twitter. How do you know when to engage, when someone is worth talking to? Melinda was very generous in helping me understand what she wrote, as was Grace Chen, so I figured I’d write a bit explaining what I understood, out of thanks.

***

About 15% of public school students are Black:

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About 16% of public school students were Black in 2012, but only 7% of teachers were Black.

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(Figures and sources from this NCES report.)

So even as Black people make up about 12% of the general population in the US and even as that share of the population is modestly rising, the percentage of Black teachers is not increasing. A disproportionate percentage of the teaching population is not Black.

The question is, why? There are roughly two sorts of answers.

Supply-side: The issue is in the pipeline. This is mostly what people talk about, and it takes myriad forms: Black people graduate high school and college at lower rates, and hence are less available for teaching positions. Black people are disproportionately uninterested in becoming teachers, for whatever reasons.

Indeed, that NCES report that I linked to focuses on the pipeline:

Screenshot 2018-01-25 at 8.46.32 PM.png

But there’s another set of factors to look at: what if the teachers are there, but schools just won’t hire Black teachers?

Demand-side: Districts are biased, either implicitly or explicitly, against Black teachers. For reasons rooted in stereotype or false-associations, Black candidates are less appealing to schools, so they don’t get hired.

That’s what Melinda wrote about in the Atlantic piece. She wrote about districts that were sued for being biased against Black teachers, and the court agreed. (Or, rather, one district was issued a decision by a court to stop discriminating against Black teachers, and another took action after they got spooked by a suit that was in the works.)

One thing I was wondering is whether it’s really possible to disentangle supply and demand considerations. If Black teachers experience bias in school hirings then they’ll be less likely to pursue careers in schooling. If you start with a dearth of Black teachers in the pipeline, biases will creep up. These factors might not be possible to truly separate.

My own school has disproportionately few non-White teachers (and students). Is the issue supply-side or demand-side? Both, probably, though I suppose it only matters to the extent that this helps a school decide what to do. As Grace points out, most in education prefer to frame this as a “won’t somebody think about the children” issue, i.e. it helps kids to have Black teachers. I’m not saying I’m skeptical of that, but as Grace notes this could be a way for schools to avoid responsibility for bias — the dearth of Black teachers is then not unlike having not enough computers or too few pencils.

But how do you tell a school to stop being biased against Black job candidates? The answer for both of these schools was affirmative-action policies. Is there a stronger case for affirmative action in the presence of bias than if the problem is in the pipeline? I think so, though I feel fuzzy about how to think about this.

Many people I talk to who are otherwise strong advocates for diversity in the teaching force are uncomfortable with affirmative action. But maybe there’s an important difference to folks between affirmative action to correct a present bias vs. affirmative action to address a historical bias?

I’m confused. Melinda said at one point in our Twitter chat that she wrote the piece to provoke thinking and discussion, and it’s definitely done that for me.

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

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In my attempt to understand the first paragraph and the notion of a topology (and a topological space) I realized that there was something I wasn’t getting. I understood the definition of “open” subsets of X, I also understood the meaning of “neighborhoods,” and I felt like I was getting a grasp on the axioms.

But what does it mean to have a “notion of nearness”?

Now, I think I’m starting to get it. In Calculus or Real Analysis you define a continuous function in terms of distance and the “I can get closer” game. But the point is that you don’t need distance in order to define continuity or a lot of those other analytic notions. All you need is a “notion of nearness,” and if open sets and neighborhoods make sense in your space, then you can usefully obtain that notion of nearness.

The usual Calculus definition of a continuous mapping goes like this: if we want all the values in the range to be within a certain neighborhood — no matter how narrow the neighborhood — we can always find a suitable neighborhood in the domain whose points all map to that band of points in the range. In this, a neighborhood is defined in terms of distance — wanting to be in a certain neighborhood in the range is being a certain (potentially tiny, epsilon) distance away from a point in the range.

But who needs to define a neighborhood in this way? We can replace this metric notion of neighborhood with a topological one, the one that we struggled to grasp in the previous post, that really comes down to open subsets. So, in short, a function is continuous at a point in a topological space if, no matter the neighborhood in the range (call it Y), there’s always a neighborhood in the domain (X) that the function maps to a neighborhood that’s a subset of the original (Y).

I should probably be trying to make these images myself rather than stealing them from wikipedia but whatever:

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So now I’m thinking of topological spaces as metric spaces, minus distance. Which explains something I never really understood, which is why in topology you can deform all sorts of things into other things. Why should a coffee mug and a doughnut be equivalent in a topological sense? The answer is that distance doesn’t exist in a topological space. All we have is open sets and subsets of open sets which stands in for our notions of nearness. As long as a mapping (like a homeomorphism) preserves the open set structure — the idea that things that start out near each other should remain near each other, relatively speaking — then there is a topological sameness that is preserved.

Honestly I should probably just be learning topology instead of trying to make sense of this algebraic topology text, but let’s see how far we can go. The next paragraph makes the jump to algebraic topology.

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:

360px-Topological_space_examples.svg.png

  • 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