More applications of winding numbers | Algebraic Topology | NJ Wildberger

We define the degree of a function from the circle to the circle, and use that to show that there is no retraction from the disk to the circle, the Brouwer fixed point theorem, and a Lemma of Borsuk.

This is the 13th lecture of this beginner's course in Algebraic Topology, given by Assoc Prof N J Wildberger at UNSW.

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[postbodzapism]

This is really a inspiring answer - I have never thought that continuity can be treated in more than one ways...
Perhaps I should wait patiently for your MathFoundations series that talks about all these...

[njwildberger]

I guess it depends on what you mean by continuous. It is a valid model of the continuum.

[pseudolullus]

Great lecture, I really like the continuous mappings

[mettikhoramshahi]

it continuous in some sense because "there is rational number between any two rational number"

[ericterry]

But isn't it disconnected (everywhere)?  If so, then what does that say about its fundamental group?  Your ideas about real numbers are fascinating.  I too have a problem with the real numbers and I have some theories/hunches.  I just can't justify them mathematically.  Would like to discuss them with you some time.