Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Must every compact and connected metric space be locally connected at at least one of its points?

share|improve this question
3  
Umm.. I just noticed that the answer to this was given more than a year ago to another question made by you on MO: mathoverflow.net/questions/36488/… –  Tapio Rajala Jan 20 '12 at 13:53

5 Answers 5

There is a "folklore" counterexample. Peter Nyikos gives the construction here (see the last paragraph for the compactness)

share|improve this answer
    
I assume it's folklore, but people more versed in this sort of thing may know differently... –  Todd Eisworth Jan 19 '12 at 21:22
3  
Nyikos' example seems fine, but I think he incorrectly uses positive slopes for the "left fan" and negative slopes for the "right fan." I see it making more sense with positive slopes in both. –  Matt Brin Jan 20 '12 at 0:08
    
It's one of those examples where the picture is much easier to comprehend than the description. –  Todd Eisworth Jan 20 '12 at 0:17
    
And yes, the picture would seem to have slopes being positive in both fans. –  Todd Eisworth Jan 20 '12 at 2:30

One more picture: Brouwer--Janiszewski--Knaster continuum:

alt text

share|improve this answer

In the product of the closet unit interval $I$ with the Cantor set $C$, identify $(0,x)$ and $\big(1,f(x)\big)$ where $f(x):=3x \mod 1$.

The resulting space $X$ is the mapping cylinder of the continuous map $f:C\to C$. It is a compact metric space locally homeomorphic to $]0,1[\times C\, ,$ thus not locally connected at any point. End-points of $I$ have not a special role; we may equivalently obtain $X$ with a larger quotient, $(\mathbb{R}\times C) / \{ (t,x)=\big (t+1,f(x)\big) \} $. The important feature of the map $f:C\to C$ is that it has a dense orbit $f^n( x_0)$. This is easily seen as it is conjugate to the left-shift map on binary strings on the space ${\bf 2}^\mathbb{N}$, $(c_1,c_2,\dots)\mapsto(c_2,c_3,\dots)$ which is just how we see $f$ on the 2-digits representations of points of the Cantor set. As a consequence, the image of $\mathbb{R}\times \{x_0\}$ in the latter quotient is a path-connected dense subset of $X$, which is therefore connected.

edit. Actually, such spaces are quite common in dynamical systems; an other example is the Smale-Williams Solenoid and several strange attractors.

share|improve this answer

The two examples given so far do not contain an illustration of the set. So here is one drawn with a "Cantor-pen":

alt text

share|improve this answer

No, examples abound, e.g., the $\sin\frac1x$-curve, i.e., the closure of $\lbrace \sin\frac1x : 0 < x \le 1\rbrace$ in the plane. As noted below, this is not a good example (I misread the question).

However, every indecomposable continuum, such as Knaster's bucket-handle continuum, is an example because every proper subcontinuum is nowhere dense. The pseudo-arc is, of course the ultimate example.

share|improve this answer
1  
not this one: it has tons of points with locally connected neighbourhoods: anything off the $y$ axis –  Anthony Quas Jan 19 '12 at 22:01
2  
Ah yes, I misread the question but I added the bucket handle just to be sure. There's also the pseudoarc of course. –  KP Hart Jan 20 '12 at 9:08
    
I am really embarassed and must apologize to everyone for failing to keep track of some of my past questions-in particular No.36488. As Tapio Rajala has pointed out, this question is almost a duplicate of my present question and the very neat answer to it given by Victor Protsak also answers my present question-which should probably be closed out on the grounds of almost duplicating a previous question. Many thanks to all of you for your excellent answers which I was never quite able to discover for myself. –  Garabed Gulbenkian Jan 21 '12 at 19:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.