A question about local connectedness in metric spaces

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

-
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

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

-
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
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

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.

-

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

-

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.

-
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
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