Among topological spaces, the metric spaces are usually considered to be the tame animals. Describing the topological notion of closeness by a distance is so intuitive (as opposed to the abstract definition of topology) that we don't expect metric spaces to be too wild. But it is such a wide class of objects that I think they can nonetheless be quite counter-intuitive and pathological in many ways.
What are your favourite, maybe eye-opening examples, preferably with properties like you wouldn't expect (not necessarily purely metric or topological properties)?
As a starting point I will list some examples which, at this point, no longer really surprise me, but they exhibit some properties which may defy (or expand) intuition:
- The plane endowed with the center metric (that is, the distance between two points is either the Euclidean distance between them if they lie on the same line through the origin, or the sum of their distances to the origin otherwise). In this example, balls around zero are just Euclidean balls, but balls around any other point look like the union of an Euclidean about origin (which may possibly be empty) and an open interval. This shows that a metric space need not look the same everywhere, and that balls can grow in a very non-uniform way.
- No matter how large a set we choose, if we endow it with discrete metric. This shows that a metrisable (even completely metrisable and locally compact) space need not be small in the sense of cardinality, and that we can put the structure of a metric space on any set.
- Further expanding on the "bigness" front, for any cardinality $\kappa$ there is a (completely metrisable) Hilbert space of dimension $\kappa$. For suitably large $\kappa$ they can be very large in many different ways, for example, any dense subset of such a space has cardinality at least $\kappa$, they can hold $\kappa$ disjoint balls. On the other hand, any compact subset of such a space (for infinite $\kappa$) has empty interior, and they are still path-connected, so very far from discrete. Related examples are general normed spaces, including operator spaces and abstract Banach spaces. I'm curious some other practical examples of very large metric spaces (which are not normed vector spaces).
- A kind of meta-example are ultrametric spaces, satisfying the strong triangle inequality (see wiki). They have the property that every triangle is isosceles, and that any open ball is centered at each of its points. Note that discrete spaces are ultrametric.
- Classical examples of ultrametric spaces are the Cantor space $2^\omega$ of binary sequences and the Baire space $\omega^\omega$ of sequences of natural numbers. They're both Polish, and even though they're perfect (every point is a limit point), they're totally disconnected. The former is compact, while the latter has the property that no compact set has nonempty interior.
- If we take a Bernstein set $X$ on the real line (that is, a set such that its intersection with any uncountable closed set is nonempty, and whose complement also has the property), then it is a separable metric space of size continuum, which has the property that every compact subset is countable. From this it follows that no nonzero continuous measure on $X$ is Radon, since every such measure must be zero on all compact sets. This places $X$ on the opposite of Polish spaces among metrizable spaces in a way, since any bounded Borel measure on a Polish space is Radon. Note that any probability Borel measure on a metric space has to be regular, so we can't get much more pathological than that. On a different front, a Bernstein set is a Baire space (the intersection of a countable family of open dense sets is dense), but the associated Banach-Mazur game is undecidable. It is, in many ways, the most pathological a subset of a Polish space can be.
- For a more geometric example, a Gromov Hyperbolic group endowed with word length metric has the property that for any triangle, any point on one side can only be away from the two other sides by a distance bounded by a constant, no matter what the vertices of the triangle might be. I think this is a property shared by hyperbolic spaces, but I'm not very familiar with hyperbolic geometry so I'd rather not say something I'm not sure about. :)