Timeline for Can the Lawvere fixed point theorem be used to prove the Brouwer fixed point theorem?

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when toggle format	what		by	license	comment
S Jun 17, 2023 at 21:09	history	suggested	Jason Gross	CC BY-SA 4.0	Fix typo, add dumb comment so that the "at least 6 characters" requirement is met
Jun 17, 2023 at 18:32	review	Suggested edits
S Jun 17, 2023 at 21:09
Apr 22, 2017 at 7:35	comment	added	Sam Eisenstat		You can't have $e$ continuous w.r.t. the sup metric and have dense image, because that would give you a continuous surjection $\beta A \to B^{\beta A}$, where $\beta A$ is the Stone-Čech compactification, since $(B^A, \sup) \cong (B^{\beta A})$. You might be able to get $e$ continuous w.r.t. the exponential topology but have dense image w.r.t. the sup metric. Also, even though @ToddTrimble retracted his answer, you can't have a continuous surjection $A \to D^A$ with $A$ compact Hausdorff and $D$ a disk because $D^A$ would have the topology of the unit ball of a Banach space.
Apr 19, 2017 at 21:36	comment	added	Andrej Bauer		We now have a bunch of results showing that the direct Lawvere fixed-point theorem is unlikely to succeed (see @ToddTrimble's answer). Can someone show that we can have $X \to [0,1]^X$ with a dense image? Or can we?
Jul 13, 2013 at 12:35	history	edited	Andrej Bauer	CC BY-SA 3.0	added 220 characters in body
Jul 13, 2013 at 8:03	history	answered	Andrej Bauer	CC BY-SA 3.0