Update: comments on this post are now closed, since my latest post would compromise any further contributions to the experiment.
Most of this post consists of write-ups of proofs of five simple propositions about metric spaces. There are three write-ups per proof, and I would be very grateful for any comments that you might have. If you would like to participate in the experiment, then please state your level of mathematical experience (the main thing I need to know is whether you yourself have studied the basic theory of metric spaces) and then make any comments/observations you wish on the write-ups. The more you say, the more useful it will be (within reason). I am particularly interested in comparisons and preferences. For each proof, the order of the three write-ups has been chosen randomly and independently.
It would also be useful if you could rate each of the 15 write-ups for clarity and style. So that everyone rates in the same way, I suggest the following rating systems.
-2 very hard to understand
-1 hard to understand
0 neither particularly hard nor particularly easy
1 easy to understand
2 very easy to understand
-2 very badly written
-1 badly written
0 neither badly written nor well written
1 well written
2 very well written
I stress that ratings should not be regarded as a substitute for comments and observations, or vice versa. What I really need is both comments and numerical ratings.
I do not want people to be influenced by the answers that other people give, so all comments on this post will go to my moderation queue. When I have enough data for the experiment, probably in a week or so, I will publish all the comments (unless for some reason you specifically request that your comment should not be published).
The more people who participate, the more reliable the results of the experiment will be. I realize that it may take a little time, so thank you very much in advance to everybody who agrees to help. (Update 26th March: I now have over 30 responses; they have been very helpful indeed, so I am extremely grateful for those. If they keep coming in at a similar rate over the next few days it will be wonderful.)
Problem 1. Let and be open sets in a metric space. Then is open.
1(a) We want to show that for all in , there exists such that the open ball is contained in .
Let . is in , so is in and is in . Since and are open, there exist such that and . Let . Then is contained in , so it’s contained in . Similarly, is contained in , so it’s contained in . So . So is open.
1(b) For arbitrary , let . Consider an arbitrary . As are open there are such that and . Take . Then and . So . We’ve proved that for any there is an open ball ( in this case) that contains and is inside . So is open.
1(c) Let be an element of . Then and . Therefore, since is open, there exists such that whenever and since is open, there exists such that whenever . We would like to find s.t. whenever . But if and only if and . We know that whenever and that whenever . Assume now that . Then if and if . We may therefore take and we are done.
Problem 2. Let and be metric spaces, let be continuous, and let be an open subset of . Then is an open subset of .
2(a) Let be an element of . Then . Therefore, since is open, there exists such that whenever . We would like to find s.t. whenever . But if and only if . We know that whenever . Since is continuous, there exists such that whenever . Therefore, setting , we are done.
2(b) Let . We seek such that the open ball is contained in .
, so . is open, so we know that for some , . Since is continuous, there exists such that for all , ; i.e., . So if ; i.e., . So is open.
2(c) Take any . We have . As is open, there is an open ball in . Because is continuous, there is some such that for any , belongs to . Hence, for such , . So . So . We’ve proved that every point in has an open ball neighbourhood. So is open.
Problem 3. Let be a complete metric space and let be a closed subset of . Then is complete.
3(a) Consider an arbitrary Cauchy sequence in . As is complete, has a limit in . Suppose . Because is closed, belongs to . We’ve proved that every Cauchy sequence in has a limit point in . So is complete.
3(b) Let be a Cauchy sequence in . Then, since is complete, we have that converges. That is, there exists such that . Since is closed in , is a sequence in and , we have that . Thus converges in and we are done.
3(c) Let be a Cauchy sequence in . We want to show that tends to a limit in .
Since is a subset of , is a Cauchy sequence in . Since is complete, , for some . Since is a closed subset of , it must contain all its limit points, so . So in . So is complete.
Problem 4. Let and be metric spaces and let and be continuous. Then the composition is continuous.
4(a) Let , and let . We need to show that there exists such that for all , .
is continuous, so there exists such that for all , . is continuous, so there exists such that for all , . But then , as desired. So is continuous.
4(b) Take an arbitrary . Let and . Using continuity of , for any , there is some such that if (for ), then . As is continuous, there is some such that if (for ), then . So for any we’ve found such that if , then and therefore . Hence is continuous.
4(c) Take and . We would like to find s.t. whenever . Since is continuous, there exists such that whenever . Since is continuous, there exists such that whenever . Therefore, setting , we are done.
Problem 5. Let and be sets, let be an injection and let and be subsets of . Then .
5(a) Take . So there is some and such that . As is injective, and are equal. So . So .
5(b) Suppose . Then, for some , and . So . Since is injective, , so , so . So .
5(c) Let be an element of . Then and . That is, there exists such that and there exists such that . Since is an injection, and , we have that . We would like to find s.t. . But if and only if and . Therefore, setting , we are done.