Question

There is a compact connected subset of R^2 which admits no nonconstant path

Answer 1

ELIphysicist?

Answer 2

Singleton set

Answer 3

There's also the pseudocircle which has similarly bizarre properties

Answer 4

For those who might find the post confusing, OP’s question to his colleagues presumably also included the qualifier that no two points in X could be connected by a smooth path.

Edit: No, it was a minor wording issue, which the OP has now fixed. See below.

Answer 5

Interesting. Do you happen to know if it’s consistent that there is a compact connected subset of the plane of cardinality <&cfr;?

Answer 6

Just to comment that the following generalization of your question that inspired this post is exercise 4 to section 27 of Munkres' *Topology* (2nd edition):

A connected metric space with more than one point is uncountable.

Proof: Fix a point p in your metric space and consider the function from your space to the real line given by distance to p. The image is connected and contains more than one point, hence is uncountable.

Answer 7

This paper is really well written for an undergraduate. Super enjoyed reading that. Ryan Wandsnider's got some chops.

I've always enjoyed this Bing style point set pathologies genre of math.

Answer 8

Wow, that's really weird. Good to remember though, like all pathological cases in topology.

Answer 9

if it's in R\^2 why can't we have an image of it

(every pixel that contains some of it is filled in)

Answer 10

Arbitrary connected subset X of R\^n with at least two points can be mapped ONTO interval \[0;1\] hence X has cardinality of the continuum.  


More generally, if X is a connected completely regular space that has at least two different points then the cardinality of X is at least continuum.