Wherein Arunas confuses the kids
Jul. 11th, 2007 05:00 pmArunas has done some confusing things before, but not quite like this. Mostly they were on the side board, where he writes when he goes off on tangents (which, contrary to my earlier assertion, usually have nothing to do with Lenin) and thus "you don't have to understand it".
Also it was mostly limited to not defining things - like talking about cell decompositions without giving any idea what they were. I mean, not formally, we wouldn't expect - these kids don't even know what a topological space is - but the informal idea of a cell decomposition of a space isn't so hard to get across (and I think I managed to do so, for the case of a sphere, anyway, which was really all we needed as he was just talking about Euler characteristic of the sphere is 2 - of course then he tried to talk about Euler characteristic in general on the side board[0].).
Today it got really bad, though. He'd been using whole numbers as sets (via their standard set theory representation) all along, for random examples, and I kind of doubt everyone understood him, but today he was really confusing with it (well, I mean, for someone who hadn't seen this before). He talked about a function f:4→3, and then proceeded to write down what it did to the elements of 4 - 0→2, 1→0... "Um, Arunas? You're using the wrong arrow."[4] Not that the kids had seen arrow-with-a-bar-on-the-left anyway (of course I explained that afterward), but we had to at least correct him so there was some distinction... of course it didn't really reduce the confusion. Thankfully the kids seemed to understand it once I explained it explicitly. (It's not like he hadn't said that 3={0,1,2} and so forth before, just that he would do it in passing whenever he wanted to use an unimportant finite set, and never made a big note of it. Which, given the potential for confusion, is pretty important.)
He also likes to invent his own notation! While he didn't try to do function composition the other way around (aw...), when we were meeting with him before YSP started to learn what we would be doing, he did something really strange: He decided he didn't like the notation f-1 for inverse image - because it's not actually an inverse - and instead proposed the completely bizarre f∈. Jonathan suggested using another notation he found in a book somewhere, f with a little -1 written vertically on top of it, as a compromise, though thankfully we eventually convinced him to just go with the standard notation there. Today, however, he surprised us by introducing new notation for intersection and union! He said, we have γ:Γ→℘(S) an indexed family of subsets of S [3], we'll define their intersection ∩γ and their union ∪γ! Huh?! "Um, Arunas? Maybe you should show them the standard notation?"[5] And so he does, and he says he likes his better because it's cleaner. Now while I certainly like the notation ∩A when A is a set of sets - it's nice and clean - when your A is a function it just looks flat-out weird and unintuitive. I'd have to go with the standard notation on that one. (Alternatively, you could use the ∩A notation by saying ∩γ(Γ), but really, probably better to introduce it first with the standard notation even if you were talking about sets of sets, as it's just more explicit.)
Unfortunately, he *did* use the standard definition of a topology. If I haven't said it here before, I'm a big fan of the Kuratowski closure operator definition, which I think is much more intuitive than the ordinary open set definition. When anyone asks me what a topology is, and they actually want more than a rough idea about donuts and coffee cups, this is the explanation I give (though actually I start instead with a proximity relation δ between X and ℘(X) which says when a point is close to a set, and then note that such a relation is equivalent to a function ℘(X)→℘(X) (as you kind of have to for the idempotence axiom)). I have actually managed to explain what a topology basically formally is to nonmath people this way, whereas I can't imagine the open set definition would do anything but confuse them. (I just think it's much more intuitive to speak of "When is a point close to a set?" than to speak of "When is a set open?" - what the hell does open mean? People say all the time that topology gives an idea of "closeness", why not define it that way?)
Anyway. He ended last time by giving the formal definition of a topology, with absolutely no examples - "just something for you to think about". (I afterward started explaining about metric spaces, to give them an idea of a more typical[6] topology, as it's kind of the motivating example. Still using the open set definition of topology, of course, as that's what he used and I'm trying to resolve confusion, not cause it.) And today he started giving examples - of course, the discrete and indiscrete (which he called concrete) topologies, and then what? Well, he starts going over finite topologies. All the topologies on ∅. All the topologies on a 1-element set. All the topologies on a 2-element set. He didn't get to 3. And then, he said, when time ran out, he had wanted to talk about the product topology, before he had even given a single nonstupid[8] example![7]
I'm looking at Munkres right now - Munkres, I remembered, also just started straight up with the open set definition, but does it a bit more sensibly. After giving the trivial examples, and showing some topologies on 3 elements, its next example is the cofinite topology. Not a very good example, no, but good when you've seen nothing else, as it quickly demonstrates a nontrivial topology on any number of points. It then defines the topology generated from a basis, and uses this to start the real examples - R, several alternate topologies on R, and also on Q. Then it talks about the order topology, and then leads into the basic general constructions - product, subspace, etc. It doesn't get to metric spaces till after this, but it still gives a good base of examples beforehand, especially with the various order topologies. (Hm, maybe I should have tried talking about just R or something like that rather than metric spaces... oh well.) As opposed to trying to explain product topology without any nonstupid examples (and without having talked about bases for topologies).
The kids have their own problems, specifically with obvious questions. Not obvious as in intuitive, but obvious as in just apply the definition. Is the following collection of subsets of {0,1} a topology? Shrug. I found out today, though, that a big part of the reason[9] that they have trouble applying the definitions is that they don't write them down. So I made a big point today that they should "use their paper memory".
But back to Arunas. He seems to have decided he's not going to say "continous function" anywhere. He's just going to say "map". Now, this works when you have people who are used to the idea that different sorts of structures have different appropriate maps - group homomorphisms, ring homomorphisms, etc... but even then you should still say "map of topologies" at first, rather than just "map". Because, you know, that kind of just means "function". And, you know, seeing as "continuous function" is the term for them, and that these kids aren't used to the idea that different sorts of structures have different appropriate morphisms - and considering this is just basic point-set topology, so you may well find yourselves dealing with functions that actually are not continuous - well... yeah.
Blech.
-Harry
[0]I should also note, he has a love for trivial examples. S0 not trivial enough for you? How about S-1, the empty set in 0 dimensional space? Obviously, it has Euler characteristic 0. And, while trivial examples are obviously very important, it's not so good when you try to use them as real examples rather than "these are the trivial ones we can get out of the way but will have to watch out for".
[3]Though of course it could be an aribtrary indexed family of sets.
[4]This was me.
[5]This was Sam.
[6]Yeah, yeah, I'm sure if you define a way of talking about "almost all topologies" your typical topology is actually completely pathological, but YKWIM.
[7]I should note here - before class began, we joked about topologies he might try to introduce but absolutely shouldn't; our examples were all general constructions that relied on having other topologies first - subspace, quotient, disjoint union... but someone mentioning the product topology was what started that line of thought.
[8]I'm not using "stupid" to mean "trivial" as Rosenberg does - I mean "stupid". The topology {∅,{0},{0,1}} on {0,1} isn't trivial, but it's stupid.
[9]Well, I assume it's a big part...
Also it was mostly limited to not defining things - like talking about cell decompositions without giving any idea what they were. I mean, not formally, we wouldn't expect - these kids don't even know what a topological space is - but the informal idea of a cell decomposition of a space isn't so hard to get across (and I think I managed to do so, for the case of a sphere, anyway, which was really all we needed as he was just talking about Euler characteristic of the sphere is 2 - of course then he tried to talk about Euler characteristic in general on the side board[0].).
Today it got really bad, though. He'd been using whole numbers as sets (via their standard set theory representation) all along, for random examples, and I kind of doubt everyone understood him, but today he was really confusing with it (well, I mean, for someone who hadn't seen this before). He talked about a function f:4→3, and then proceeded to write down what it did to the elements of 4 - 0→2, 1→0... "Um, Arunas? You're using the wrong arrow."[4] Not that the kids had seen arrow-with-a-bar-on-the-left anyway (of course I explained that afterward), but we had to at least correct him so there was some distinction... of course it didn't really reduce the confusion. Thankfully the kids seemed to understand it once I explained it explicitly. (It's not like he hadn't said that 3={0,1,2} and so forth before, just that he would do it in passing whenever he wanted to use an unimportant finite set, and never made a big note of it. Which, given the potential for confusion, is pretty important.)
He also likes to invent his own notation! While he didn't try to do function composition the other way around (aw...), when we were meeting with him before YSP started to learn what we would be doing, he did something really strange: He decided he didn't like the notation f-1 for inverse image - because it's not actually an inverse - and instead proposed the completely bizarre f∈. Jonathan suggested using another notation he found in a book somewhere, f with a little -1 written vertically on top of it, as a compromise, though thankfully we eventually convinced him to just go with the standard notation there. Today, however, he surprised us by introducing new notation for intersection and union! He said, we have γ:Γ→℘(S) an indexed family of subsets of S [3], we'll define their intersection ∩γ and their union ∪γ! Huh?! "Um, Arunas? Maybe you should show them the standard notation?"[5] And so he does, and he says he likes his better because it's cleaner. Now while I certainly like the notation ∩A when A is a set of sets - it's nice and clean - when your A is a function it just looks flat-out weird and unintuitive. I'd have to go with the standard notation on that one. (Alternatively, you could use the ∩A notation by saying ∩γ(Γ), but really, probably better to introduce it first with the standard notation even if you were talking about sets of sets, as it's just more explicit.)
Unfortunately, he *did* use the standard definition of a topology. If I haven't said it here before, I'm a big fan of the Kuratowski closure operator definition, which I think is much more intuitive than the ordinary open set definition. When anyone asks me what a topology is, and they actually want more than a rough idea about donuts and coffee cups, this is the explanation I give (though actually I start instead with a proximity relation δ between X and ℘(X) which says when a point is close to a set, and then note that such a relation is equivalent to a function ℘(X)→℘(X) (as you kind of have to for the idempotence axiom)). I have actually managed to explain what a topology basically formally is to nonmath people this way, whereas I can't imagine the open set definition would do anything but confuse them. (I just think it's much more intuitive to speak of "When is a point close to a set?" than to speak of "When is a set open?" - what the hell does open mean? People say all the time that topology gives an idea of "closeness", why not define it that way?)
Anyway. He ended last time by giving the formal definition of a topology, with absolutely no examples - "just something for you to think about". (I afterward started explaining about metric spaces, to give them an idea of a more typical[6] topology, as it's kind of the motivating example. Still using the open set definition of topology, of course, as that's what he used and I'm trying to resolve confusion, not cause it.) And today he started giving examples - of course, the discrete and indiscrete (which he called concrete) topologies, and then what? Well, he starts going over finite topologies. All the topologies on ∅. All the topologies on a 1-element set. All the topologies on a 2-element set. He didn't get to 3. And then, he said, when time ran out, he had wanted to talk about the product topology, before he had even given a single nonstupid[8] example![7]
I'm looking at Munkres right now - Munkres, I remembered, also just started straight up with the open set definition, but does it a bit more sensibly. After giving the trivial examples, and showing some topologies on 3 elements, its next example is the cofinite topology. Not a very good example, no, but good when you've seen nothing else, as it quickly demonstrates a nontrivial topology on any number of points. It then defines the topology generated from a basis, and uses this to start the real examples - R, several alternate topologies on R, and also on Q. Then it talks about the order topology, and then leads into the basic general constructions - product, subspace, etc. It doesn't get to metric spaces till after this, but it still gives a good base of examples beforehand, especially with the various order topologies. (Hm, maybe I should have tried talking about just R or something like that rather than metric spaces... oh well.) As opposed to trying to explain product topology without any nonstupid examples (and without having talked about bases for topologies).
The kids have their own problems, specifically with obvious questions. Not obvious as in intuitive, but obvious as in just apply the definition. Is the following collection of subsets of {0,1} a topology? Shrug. I found out today, though, that a big part of the reason[9] that they have trouble applying the definitions is that they don't write them down. So I made a big point today that they should "use their paper memory".
But back to Arunas. He seems to have decided he's not going to say "continous function" anywhere. He's just going to say "map". Now, this works when you have people who are used to the idea that different sorts of structures have different appropriate maps - group homomorphisms, ring homomorphisms, etc... but even then you should still say "map of topologies" at first, rather than just "map". Because, you know, that kind of just means "function". And, you know, seeing as "continuous function" is the term for them, and that these kids aren't used to the idea that different sorts of structures have different appropriate morphisms - and considering this is just basic point-set topology, so you may well find yourselves dealing with functions that actually are not continuous - well... yeah.
Blech.
-Harry
[0]I should also note, he has a love for trivial examples. S0 not trivial enough for you? How about S-1, the empty set in 0 dimensional space? Obviously, it has Euler characteristic 0. And, while trivial examples are obviously very important, it's not so good when you try to use them as real examples rather than "these are the trivial ones we can get out of the way but will have to watch out for".
[3]Though of course it could be an aribtrary indexed family of sets.
[4]This was me.
[5]This was Sam.
[6]Yeah, yeah, I'm sure if you define a way of talking about "almost all topologies" your typical topology is actually completely pathological, but YKWIM.
[7]I should note here - before class began, we joked about topologies he might try to introduce but absolutely shouldn't; our examples were all general constructions that relied on having other topologies first - subspace, quotient, disjoint union... but someone mentioning the product topology was what started that line of thought.
[8]I'm not using "stupid" to mean "trivial" as Rosenberg does - I mean "stupid". The topology {∅,{0},{0,1}} on {0,1} isn't trivial, but it's stupid.
[9]Well, I assume it's a big part...
