M. Gavrilovich, Point-set topology as diagram chasing computations, The De Morgan Gazette 5 no. 4 (2014), 23-32.
We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms \(T_0\) and \(T_1\) in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain).
We also offer a couple of brief speculations on cognitive and AI aspects of this observation, particularly that in point-set topology some arguments read as diagram chasing computations with finite preorders.
An experimental MOOC (Massive Open Online Course) Citizen Maths is launched and the first phase of the course is open for registration. It is free and open for everyone; its motto is Powerful Ideas in Action.
The readers of this blog may like to register for the course, because, as the organisers say,
The success of this first phase of Citizen Maths will depend crucially on the feedback that we obtain. We are particularly keen to get feedback from:
- learners who do the course;
- those with an interest in the learning and teaching of maths, and in the design of online courses.
There is a link to a feedback form on every page of the Citizen Maths web site, and there will be a similar link on every page of the course when it goes live on or around 12 September.
The first pilot stage will run for four weeks and cover the first “powerful idea”: proportion. An admirable choice (a detailed discussion of the role of proportions in elementary mathematics can be found in this paper by Tony Gardiner).