I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it would be nice to have a list of such examples! (I dug through the internet without locating such a collection.)
This question/request can be interpreted as either
1) An example that obeys a particular definition of dimension but goes against our intuition. Said differently: an example that should obey a particular definition of dimension, but doesn't.
3) An example that disagrees with two different definitions of dimension.
4) An example which hinges on a hypothesis of the dimension.
*This last one is what got me to start this post, because I came across an example involving the Krull dimension: If our ring $R$ is Noetherian then $\dim R[x]=1+\dim R$, but if $R$ is not Noetherian then we can have $\dim R[x]=2+\dim R$. Found at http://www.jstor.org/stable/2373549?origin=crossref (The Dimension Sequence of a Commutative Ring, by Gilmer).
*I am not sure where our space-filling curves fit in here.
Some standard definitions of dimension
- Lebesgue covering dimension (of a topological space)
- Cohomological dimension (of a topological space)
- Hausdorff dimension (of a metric space)
- Krull dimension (of a ring or module)
Best Answer
Erdős space, the set of all vectors in $\ell^2$ with rational entries, seems like it would fit the bill -- it is a metrizable space which has "dimension one", but it is homeomorphic to its Cartesian square, and so violates our hope/intuition that $\dim(E\times F)=\dim(E)+\dim(F)$.
See Gerald Edgar's answer to a previous MO question.
(Digression: I learned of this example in a seminar given here by a postdoc, and realized as she was writing down these properties that I'd actually seen it mentioned -- without any of the relevant technical detail -- in one of the pop-maths biographies of Erdős. The story goes that he got interested in something two topologists were trying to figure out, got fobbed off with a quick explanation of the problem, came back to ask what a Hilbert space was, went away, and then came back to show that this space had dimension $1$ rather than the expected $0$ or $\infty$.)